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<title>Left Termination proof of ../tpdb/LP/terminweb/old-terminweb/mergesort.pl</title>
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<BR><B>Left Termination</B> of the query pattern
mergesort_in_2(g, a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><BR>Clauses:<BR><BR>mergesort([], []).<BR>mergesort(.(X, []), .(X, [])).<BR>mergesort(.(X, .(Y, Xs)), Ys)&#160;:-&#160;','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))).<BR>split([], [], []).<BR>split(.(X, Xs), .(X, Ys), Zs)&#160;:-&#160;split(Xs, Zs, Ys).<BR>merge([], Xs, Xs).<BR>merge(Xs, [], Xs).<BR>merge(.(X, Xs), .(Y, Ys), .(X, Zs))&#160;:-&#160;','(=<(X, Y), merge(Xs, .(Y, Ys), Zs)).<BR>merge(.(X, Xs), .(Y, Ys), .(Y, Zs))&#160;:-&#160;','(>(X, Y), merge(.(X, Xs), Ys, Zs)).<BR><BR>Queries:<BR><BR>mergesort(g,a).<BR><BR>Added definitions of predefined predicates.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 <B>Prolog</B></pre><pre>      &#8627 OrTransformerProof</pre><BR>Clauses:<BR><BR>mergesort([], []).<BR>mergesort(.(X, []), .(X, [])).<BR>mergesort(.(X, .(Y, Xs)), Ys)&#160;:-&#160;','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))).<BR>split([], [], []).<BR>split(.(X, Xs), .(X, Ys), Zs)&#160;:-&#160;split(Xs, Zs, Ys).<BR>merge([], Xs, Xs).<BR>merge(Xs, [], Xs).<BR>merge(.(X, Xs), .(Y, Ys), .(X, Zs))&#160;:-&#160;','(','(=(X1, X), ','(=(X2, Y), ;(=(X1, X2), isLess(X1, X2)))), merge(Xs, .(Y, Ys), Zs)).<BR>merge(.(X, Xs), .(Y, Ys), .(Y, Zs))&#160;:-&#160;','(','(=(X1, X), ','(=(X2, Y), isGreater(X1, X2))), merge(.(X, Xs), Ys, Zs)).<BR>isPlus(zero, X, X).<BR>isPlus(succ(X), zero, succ(X)).<BR>isPlus(succ(X), succ(Y), succ(succ(Z)))&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(succ(X), pred(Y), Z)&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(pred(X), zero, pred(X)).<BR>isPlus(pred(X), succ(Y), Z)&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(pred(X), pred(Y), pred(pred(Z)))&#160;:-&#160;isPlus(X, Y, Z).<BR>isMinus(X, zero, X).<BR>isMinus(zero, succ(Y), pred(Z))&#160;:-&#160;isMinus(zero, Y, Z).<BR>isMinus(zero, pred(Y), succ(Z))&#160;:-&#160;isMinus(zero, Y, Z).<BR>isMinus(succ(X), succ(Y), Z)&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(succ(X), pred(Y), succ(succ(Z)))&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(pred(X), succ(Y), pred(pred(Z)))&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(pred(X), pred(Y), Z)&#160;:-&#160;isMinus(X, Y, Z).<BR>isTimes(X, zero, zero).<BR>isTimes(zero, succ(Y), zero).<BR>isTimes(zero, pred(Y), zero).<BR>isTimes(succ(X), succ(Y), Z)&#160;:-&#160;','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).<BR>isTimes(succ(X), pred(Y), Z)&#160;:-&#160;','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).<BR>isTimes(pred(X), succ(Y), Z)&#160;:-&#160;','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).<BR>isTimes(pred(X), pred(Y), Z)&#160;:-&#160;','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).<BR>isDiv(zero, succ(Y), zero).<BR>isDiv(zero, pred(Y), zero).<BR>isDiv(succ(X), succ(Y), zero)&#160;:-&#160;isMinus(succ(X), succ(Y), pred(Z)).<BR>isDiv(succ(X), succ(Y), succ(Z))&#160;:-&#160;','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).<BR>isDiv(succ(X), pred(Y), Z)&#160;:-&#160;','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).<BR>isDiv(pred(X), pred(Y), zero)&#160;:-&#160;isMinus(pred(X), pred(Y), succ(Z)).<BR>isDiv(pred(X), pred(Y), succ(Z))&#160;:-&#160;','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).<BR>isDiv(pred(X), succ(Y), Z)&#160;:-&#160;','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).<BR>isModulo(X, Y, Z)&#160;:-&#160;','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).<BR>=(X, X).<BR>isGreater(succ(X), zero).<BR>isGreater(succ(X), pred(Y)).<BR>isGreater(succ(X), succ(Y))&#160;:-&#160;isGreater(X, Y).<BR>isGreater(zero, pred(Y)).<BR>isGreater(pred(X), pred(Y))&#160;:-&#160;isGreater(X, Y).<BR>isLess(pred(X), zero).<BR>isLess(pred(X), succ(Y)).<BR>isLess(pred(X), pred(Y))&#160;:-&#160;isLess(X, Y).<BR>isLess(zero, succ(Y)).<BR>isLess(succ(X), succ(Y))&#160;:-&#160;isLess(X, Y).<BR><BR>Queries:<BR><BR>mergesort(g,a).<BR><BR>Eliminated all or-constructs.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><pre>        &#8627 <B>Prolog</B></pre><pre>          &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>mergesort([], []).<BR>mergesort(.(X, []), .(X, [])).<BR>mergesort(.(X, .(Y, Xs)), Ys)&#160;:-&#160;','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))).<BR>split([], [], []).<BR>split(.(X, Xs), .(X, Ys), Zs)&#160;:-&#160;split(Xs, Zs, Ys).<BR>merge([], Xs, Xs).<BR>merge(Xs, [], Xs).<BR>merge(.(X, Xs), .(Y, Ys), .(Y, Zs))&#160;:-&#160;','(=(X1, X), ','(=(X2, Y), ','(isGreater(X1, X2), merge(.(X, Xs), Ys, Zs)))).<BR>isPlus(zero, X, X).<BR>isPlus(succ(X), zero, succ(X)).<BR>isPlus(succ(X), succ(Y), succ(succ(Z)))&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(succ(X), pred(Y), Z)&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(pred(X), zero, pred(X)).<BR>isPlus(pred(X), succ(Y), Z)&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(pred(X), pred(Y), pred(pred(Z)))&#160;:-&#160;isPlus(X, Y, Z).<BR>isMinus(X, zero, X).<BR>isMinus(zero, succ(Y), pred(Z))&#160;:-&#160;isMinus(zero, Y, Z).<BR>isMinus(zero, pred(Y), succ(Z))&#160;:-&#160;isMinus(zero, Y, Z).<BR>isMinus(succ(X), succ(Y), Z)&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(succ(X), pred(Y), succ(succ(Z)))&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(pred(X), succ(Y), pred(pred(Z)))&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(pred(X), pred(Y), Z)&#160;:-&#160;isMinus(X, Y, Z).<BR>isTimes(X, zero, zero).<BR>isTimes(zero, succ(Y), zero).<BR>isTimes(zero, pred(Y), zero).<BR>isTimes(succ(X), succ(Y), Z)&#160;:-&#160;','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).<BR>isTimes(succ(X), pred(Y), Z)&#160;:-&#160;','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).<BR>isTimes(pred(X), succ(Y), Z)&#160;:-&#160;','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).<BR>isTimes(pred(X), pred(Y), Z)&#160;:-&#160;','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).<BR>isDiv(zero, succ(Y), zero).<BR>isDiv(zero, pred(Y), zero).<BR>isDiv(succ(X), succ(Y), zero)&#160;:-&#160;isMinus(succ(X), succ(Y), pred(Z)).<BR>isDiv(succ(X), succ(Y), succ(Z))&#160;:-&#160;','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).<BR>isDiv(succ(X), pred(Y), Z)&#160;:-&#160;','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).<BR>isDiv(pred(X), pred(Y), zero)&#160;:-&#160;isMinus(pred(X), pred(Y), succ(Z)).<BR>isDiv(pred(X), pred(Y), succ(Z))&#160;:-&#160;','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).<BR>isDiv(pred(X), succ(Y), Z)&#160;:-&#160;','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).<BR>isModulo(X, Y, Z)&#160;:-&#160;','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).<BR>=(X, X).<BR>isGreater(succ(X), zero).<BR>isGreater(succ(X), pred(Y)).<BR>isGreater(succ(X), succ(Y))&#160;:-&#160;isGreater(X, Y).<BR>isGreater(zero, pred(Y)).<BR>isGreater(pred(X), pred(Y))&#160;:-&#160;isGreater(X, Y).<BR>isLess(pred(X), zero).<BR>isLess(pred(X), succ(Y)).<BR>isLess(pred(X), pred(Y))&#160;:-&#160;isLess(X, Y).<BR>isLess(zero, succ(Y)).<BR>isLess(succ(X), succ(Y))&#160;:-&#160;isLess(X, Y).<BR>merge(.(X, Xs), .(Y, Ys), .(X, Zs))&#160;:-&#160;','(=(X1, X), ','(=(X2, Y), ','(=(X1, X2), merge(Xs, .(Y, Ys), Zs)))).<BR>merge(.(X, Xs), .(Y, Ys), .(X, Zs))&#160;:-&#160;','(=(X1, X), ','(=(X2, Y), ','(isLess(X1, X2), merge(Xs, .(Y, Ys), Zs)))).<BR><BR>Queries:<BR><BR>mergesort(g,a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>mergesort_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>split_in</font>: (b,f,f)
<BR><FONT COLOR=#0000cc>merge_in</font>: (b,b,f)
<BR><FONT COLOR=#0000cc>isGreater_in</font>: (b,b)
<BR><FONT COLOR=#0000cc>isLess_in</font>: (b,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>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</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>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</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>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U48_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</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>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</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>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U48_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_GG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</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>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</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>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U48_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U48_GGA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>=_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U43_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GGA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U44_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U45_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U46_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U52_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U7_GGA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U50_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_GG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</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>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</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>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U48_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U48_GGA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>=_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U43_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GGA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U44_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U45_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U46_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U52_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U7_GGA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U50_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 5 SCCs with 19 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 <B>PiDP</B></pre><pre>                        &#8627 UsableRulesProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 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>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</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>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</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>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U48_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 <B>PiDP</B></pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 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>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Pi is empty.<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 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 PiDP</pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 QDPSizeChangeProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 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>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI>
<LI><FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 <B>PiDP</B></pre><pre>                        &#8627 UsableRulesProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 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>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</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>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</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>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U48_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 <B>PiDP</B></pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 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>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Pi is empty.<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 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 PiDP</pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 QDPSizeChangeProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 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>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI>
<LI><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 <B>PiDP</B></pre><pre>                        &#8627 UsableRulesProof</pre><pre>                      &#8627 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>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</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>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</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>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U48_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U48_GGA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U7_GGA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 <B>PiDP</B></pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                      &#8627 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>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><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>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U48_GGA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U7_GGA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</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>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>)) at position [5] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 Rewriting</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>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>)) at position [5] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 <B>QDP</B></pre><pre>                                        &#8627 Rewriting</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>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>)) at position [4] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 <B>QDP</B></pre><pre>                                            &#8627 Rewriting</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>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>)) at position [4] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 <B>QDP</B></pre><pre>                                                &#8627 UsableRulesProof</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>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 UsableRulesProof</pre><pre>                                                  &#8627 <B>QDP</B></pre><pre>                                                    &#8627 QReductionProof</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>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 UsableRulesProof</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 QReductionProof</pre><pre>                                                      &#8627 <B>QDP</B></pre><pre>                                                        &#8627 Instantiation</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>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 UsableRulesProof</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 QReductionProof</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Instantiation</pre><pre>                                                          &#8627 <B>QDP</B></pre><pre>                                                            &#8627 Instantiation</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>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#cc0000>z2</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 UsableRulesProof</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 QReductionProof</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Instantiation</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Instantiation</pre><pre>                                                              &#8627 <B>QDP</B></pre><pre>                                                                &#8627 Instantiation</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>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>Y</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 UsableRulesProof</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 QReductionProof</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Instantiation</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Instantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Instantiation</pre><pre>                                                                  &#8627 <B>QDP</B></pre><pre>                                                                    &#8627 Instantiation</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>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 UsableRulesProof</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 QReductionProof</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Instantiation</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Instantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Instantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Instantiation</pre><pre>                                                                      &#8627 <B>QDP</B></pre><pre>                                                                        &#8627 Instantiation</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>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 UsableRulesProof</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 QReductionProof</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Instantiation</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Instantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Instantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Instantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Instantiation</pre><pre>                                                                          &#8627 <B>QDP</B></pre><pre>                                                                            &#8627 Instantiation</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>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#cc0000>z2</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 UsableRulesProof</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 QReductionProof</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Instantiation</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Instantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Instantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Instantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Instantiation</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Instantiation</pre><pre>                                                                              &#8627 <B>QDP</B></pre><pre>                                                                                &#8627 QDPOrderProof</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>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>=_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>=_out_ag</font>(x<SUB>1</SUB>)</B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>=_out_gg</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U43_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U44_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U45_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U46_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U47_GGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 1 + x<SUB>2</SUB> + x<SUB>4</SUB> + x<SUB>5</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U48_GGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>, x<SUB>6</SUB>)</B>) = 1 + x<SUB>2</SUB> + x<SUB>4</SUB> + x<SUB>6</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U49_GGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 1 + x<SUB>2</SUB> + x<SUB>4</SUB> + x<SUB>5</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U51_GGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 1 + x<SUB>2</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_GGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 1 + x<SUB>2</SUB> + x<SUB>4</SUB> + x<SUB>5</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_GGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>, x<SUB>6</SUB>)</B>) = 1 + x<SUB>2</SUB> + x<SUB>4</SUB> + x<SUB>6</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_GGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 1 + x<SUB>2</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isGreater_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isGreater_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isLess_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isLess_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>pred</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>succ</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>zero</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 UsableRulesProof</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 QReductionProof</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Instantiation</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Instantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Instantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Instantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Instantiation</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Instantiation</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 QDPOrderProof</pre><pre>                                                                                  &#8627 <B>QDP</B></pre><pre>                                                                                    &#8627 DependencyGraphProof</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>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U48_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>)) &#8594; <FONT COLOR=#0000cc>U49_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z2</font>))
<BR><FONT COLOR=#0000cc>U51_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MERGE_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 7 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 <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>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</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>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</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>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U48_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SPLIT_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 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 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>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</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>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SPLIT_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 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 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>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>Xs</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>SPLIT_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>SPLIT_IN_GAA</font>(<FONT COLOR=#cc0000>Xs</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 <B>PiDP</B></pre><pre>                        &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</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>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</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>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U48_gga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_gg</font><BR>
<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isLess_out_gg</font><BR>
<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x5</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 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) at position [0,1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 Narrowing</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 <B>QDP</B></pre><pre>                                        &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>))))) at position [1,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)))))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 <B>QDP</B></pre><pre>                                            &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>))))) at position [1,0,1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 <B>QDP</B></pre><pre>                                                &#8627 Narrowing</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>)))) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 <B>QDP</B></pre><pre>                                                    &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) at position [0,1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 <B>QDP</B></pre><pre>                                                        &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 <B>QDP</B></pre><pre>                                                            &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X1s</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 <B>QDP</B></pre><pre>                                                                &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#cc0000>X2s</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 <B>QDP</B></pre><pre>                                                                    &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>))) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 <B>QDP</B></pre><pre>                                                                        &#8627 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>[]</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 DependencyGraphProof</pre><pre>                                                                          &#8627 <B>QDP</B></pre><pre>                                                                            &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 DependencyGraphProof</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                              &#8627 <B>QDP</B></pre><pre>                                                                                &#8627 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 DependencyGraphProof</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 DependencyGraphProof</pre><pre>                                                                                  &#8627 <B>QDP</B></pre><pre>                                                                                    &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#cc0000>y1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>))))) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 DependencyGraphProof</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 DependencyGraphProof</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                      &#8627 <B>QDP</B></pre><pre>                                                                                        &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x2</font>)))))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>=_in_ag</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>=_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>=_out_ag</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>=_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U43_gg</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U44_gg</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U45_gg</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U46_gg</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U47_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U48_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>, x<SUB>6</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U49_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U50_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U51_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U52_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_gaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>, x<SUB>6</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>[]</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isGreater_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isGreater_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isLess_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isLess_out_gg</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>merge_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>merge_out_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>mergesort_in_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>mergesort_out_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>pred</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>split_in_gaa</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>split_out_gaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>succ</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>zero</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 DependencyGraphProof</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 DependencyGraphProof</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 QDPOrderProof</pre><pre>                                                                                          &#8627 <B>QDP</B></pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 DependencyGraphProof</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 DependencyGraphProof</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 QDPOrderProof</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 <B>QDP</B></pre><pre>                                                                                                &#8627 UsableRulesProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X1s</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y1s</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Y2s</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Y1s</font>, <FONT COLOR=#cc0000>Y2s</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_gg</font>
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)))
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>merge_out_gga</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>mergesort_out_ga</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 DependencyGraphProof</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 DependencyGraphProof</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 QDPOrderProof</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 UsableRulesProof</pre><pre>                                                                                                  &#8627 <B>QDP</B></pre><pre>                                                                                                    &#8627 QReductionProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>mergesort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>merge_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U43_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U47_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U48_gga</font>(<FONT COLOR=#cc0000>x0</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>)
<BR><FONT COLOR=#0000cc>=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U49_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U50_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U46_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U51_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U52_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 DependencyGraphProof</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 DependencyGraphProof</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 QDPOrderProof</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 UsableRulesProof</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 QReductionProof</pre><pre>                                                                                                      &#8627 <B>QDP</B></pre><pre>                                                                                                        &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))), <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y_2</font>, <FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y0</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)))))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR>none<BR>Used ordering:  Matrix interpretation [3]:
<BR>Non-tuple symbols: <BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>U5_gaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)<B> )</B> = </td><td><table><tr><td>/</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>/</td></tr>
</table></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>0</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>1</td><td>1</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>2</SUB></B></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>split_out_gaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)<B> )</B> = </td><td><table><tr><td>/</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>1</td><td>/</td></tr>
</table></td><td>+</td><td><table><tr><td>/</td><td>1</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>1</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>1</td><td>1</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>2</SUB></B></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>[]</font><B> )</B> = </td><td><table><tr><td>/</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>/</td></tr>
</table></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)<B> )</B> = </td><td><table><tr><td>/</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>/</td></tr>
</table></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>0</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td><td>+</td><td><table><tr><td>/</td><td>1</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>1</td><td>1</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>2</SUB></B></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>split_in_gaa</font>(x<SUB>1</SUB>)<B> )</B> = </td><td><table><tr><td>/</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>1</td><td>/</td></tr>
</table></td><td>+</td><td><table><tr><td>/</td><td>1</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>1</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td></tr>
</table><BR>Tuple symbols: <BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>MERGESORT_IN_GA</font>(x<SUB>1</SUB>)<B> )</B> = </td><td>0</td><td>+</td><td><table><tr><td>[</td><td>0,</td><td>1</td><td>]</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>U1_GA</font>(x<SUB>1</SUB>)<B> )</B> = </td><td>0</td><td>+</td><td><table><tr><td>[</td><td>1,</td><td>0</td><td>]</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td></tr>
</table><BR><BR>Matrix type: <BR>We used a basic matrix type which is not further parametrizeable.<BR><BR><BR>As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.<BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 OrTransformerProof</pre><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 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Narrowing</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Rewriting</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Rewriting</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Narrowing</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Rewriting</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 ForwardInstantiation</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 ForwardInstantiation</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 DependencyGraphProof</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 DependencyGraphProof</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 QDPOrderProof</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 UsableRulesProof</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 QReductionProof</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 QDPOrderProof</pre><pre>                                                                                                          &#8627 <B>QDP</B></pre><pre>                                                                                                            &#8627 PisEmptyProof</pre><BR>Q DP problem:<BR>P is empty.<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U5_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>split_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>split_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_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>

