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<title>Left Termination proof of ../tpdb/LP/talp/maria/qsortapp.pl</title>
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<BR><B>Left Termination</B> of the query pattern
qsort_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>qsort(.(X, L), R)&#160;:-&#160;','(partition(L, X, L1, L2), ','(qsort(L2, R2), ','(qsort(L1, R1), append(R1, .(X, R2), R)))).<BR>qsort([], []).<BR>partition([], _, [], []).<BR>partition(.(E, R), C, .(E, Left1), Right)&#160;:-&#160;','(<(E, C), ','(!, partition(R, C, Left1, Right))).<BR>partition(.(E, R), C, Left, .(E, Right1))&#160;:-&#160;partition(R, C, Left, Right1).<BR>append([], X, X).<BR>append(.(H, X), Y, .(H, Z))&#160;:-&#160;append(X, Y, Z).<BR><BR>Queries:<BR><BR>qsort(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 CutEliminatorProof</pre><BR>Clauses:<BR><BR>qsort(.(X, L), R)&#160;:-&#160;','(partition(L, X, L1, L2), ','(qsort(L2, R2), ','(qsort(L1, R1), append(R1, .(X, R2), R)))).<BR>qsort([], []).<BR>partition([], _, [], []).<BR>partition(.(E, R), C, .(E, Left1), Right)&#160;:-&#160;','(','(=(X, E), ','(=(X1, C), isLess(X, X1))), ','(!, partition(R, C, Left1, Right))).<BR>partition(.(E, R), C, Left, .(E, Right1))&#160;:-&#160;partition(R, C, Left, Right1).<BR>append([], X, X).<BR>append(.(H, X), Y, .(H, Z))&#160;:-&#160;append(X, Y, Z).<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>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>qsort(g,a).<BR><BR>Eliminated all cuts by simply ignoring them.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 <B>Prolog</B></pre><pre>          &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>qsort(.(X, L), R)&#160;:-&#160;','(partition(L, X, L1, L2), ','(qsort(L2, R2), ','(qsort(L1, R1), append(R1, .(X, R2), R)))).<BR>qsort([], []).<BR>partition([], X, [], []).<BR>partition(.(E, R), C, .(E, Left1), Right)&#160;:-&#160;','(=(X, E), ','(=(X1, C), ','(isLess(X, X1), partition(R, C, Left1, Right)))).<BR>partition(.(E, R), C, Left, .(E, Right1))&#160;:-&#160;partition(R, C, Left, Right1).<BR>append([], X, X).<BR>append(.(H, X), Y, .(H, Z))&#160;:-&#160;append(X, Y, Z).<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>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>qsort(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>qsort_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>partition_in</font>: (b,b,f,f)
<BR><FONT COLOR=#0000cc>isLess_in</font>: (b,b)
<BR><FONT COLOR=#0000cc>append_in</font>: (b,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>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>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>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>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>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_ggaa</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>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>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>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>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>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_ggaa</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)
<BR><FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>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>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>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>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>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_ggaa</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</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>U8_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U9_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GGAA</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>
<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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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>QSORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)
<BR><FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>ISLESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>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>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>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>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>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_ggaa</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</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>U8_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U9_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GGAA</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>
<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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 4 SCCs with 12 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>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>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>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>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>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_ggaa</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</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 CutEliminatorProof</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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</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>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</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>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</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 CutEliminatorProof</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><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>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>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>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>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>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>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_ggaa</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</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 CutEliminatorProof</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><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 CutEliminatorProof</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><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 CutEliminatorProof</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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)
<BR><FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>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>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>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>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>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_ggaa</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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 CutEliminatorProof</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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)
<BR><FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><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>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>U44_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>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>U44_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>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>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>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>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>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>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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 CutEliminatorProof</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 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>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>E</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><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>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>U44_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>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U44_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></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isLess_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>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><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>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))<BR>The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3<P></LI>
<LI><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>E</font>))<BR>The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3<P></LI>
<LI><FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>)<BR>The graph contains the following edges 1 > 1, 2 >= 2<P></LI>
<LI><FONT COLOR=#0000cc>U7_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>PARTITION_IN_GGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>)<BR>The graph contains the following edges 2 >= 1, 3 >= 2<P></LI>
<LI><FONT COLOR=#0000cc>U5_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGAA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>C</font>))<BR>The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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 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>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>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>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>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>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>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>), <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>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>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_ggaa</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</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 CutEliminatorProof</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 PiDPToQDPProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L1</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>U44_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</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>U5_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U6_ggaa</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>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_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>QSORT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L1</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L2</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>=_out_ag</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>QSORT_IN_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U10_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_ga</font>(x<SUB>1</SUB>, x<SUB>2</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>, x<SUB>3</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</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>, x<SUB>3</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>U4_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_ggaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_ggaa</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><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_ggaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_ggaa</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>U9_ggaa</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>[]</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>append_in_gga</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>append_out_gga</font>(x<SUB>1</SUB>)</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>partition_in_ggaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>partition_out_ggaa</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>pred</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>qsort_in_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>qsort_out_ga</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>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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 PiDPToQDPProof</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=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L1</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>QSORT_IN_GA</font>(<FONT COLOR=#cc0000>L2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>E</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>U5_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>U44_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>U45_gg</font>(<FONT COLOR=#0000cc>isLess_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>U44_gg</font>(<FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isLess_out_gg</font>
<BR><FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isLess_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_in_ggaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Left1</font>), <FONT COLOR=#cc0000>Right</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>partition_out_ggaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>R2</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>, <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>R1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>R1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R2</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>qsort_out_ga</font>(<FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>qsort_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>partition_in_ggaa</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>U5_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U6_ggaa</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>isLess_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U45_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U44_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U9_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_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 0 SCCs with 3 less nodes.<BR><BR></body>

