YES
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<title>Left Termination proof of ../tpdb/LP/talp/maria/hanoiapp.pl</title>
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<BR><B>Left Termination</B> of the query pattern
shanoi_in_5(g, g, g, 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>shanoi(1, A, B, C, .(mv(A, C), [])).<BR>shanoi(N, A, B, C, M)&#160;:-&#160;','(>(N, 1), ','(is(N1, -(N, 1)), ','(shanoi(N1, A, C, B, M1), ','(shanoi(N1, B, A, C, M2), ','(append(M1, .(mv(A, C), []), T), append(T, M2, M)))))).<BR>append([], L, L).<BR>append(.(H, L), L1, .(H, R))&#160;:-&#160;append(L, L1, R).<BR>demo&#160;:-&#160;','(=(Size, 15), ','(format('Towers of Hanoi for ~w elements', .(Size, [])), ','(time(_, A), ','(nl, ','(nl, ','(shanoi(Size, a, b, c, Sol), ','(time(T, Sol), ','(format('The solution is: ~w', .(Sol, [])), ','(nl, ','(format('Solved in ~w msec.', .(T, [])), nl)))))))))).<BR><BR>Queries:<BR><BR>shanoi(g,g,g,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 UndefinedPredicateHandlerProof</pre><BR>Clauses:<BR><BR>shanoi(succ(zero), A, B, C, .(mv(A, C), [])).<BR>shanoi(N, A, B, C, M)&#160;:-&#160;','(','(=(X, N), ','(=(X1, succ(zero)), isGreater(X, X1))), ','(','(isMinus(N, succ(zero), U), =(N1, U)), ','(shanoi(N1, A, C, B, M1), ','(shanoi(N1, B, A, C, M2), ','(append(M1, .(mv(A, C), []), T), append(T, M2, M)))))).<BR>append([], L, L).<BR>append(.(H, L), L1, .(H, R))&#160;:-&#160;append(L, L1, R).<BR>demo&#160;:-&#160;','(=(Size, succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(zero)))))))))))))))), ','(format('Towers of Hanoi for ~w elements', .(Size, [])), ','(time(_, A), ','(nl, ','(nl, ','(shanoi(Size, a, b, c, Sol), ','(time(T, Sol), ','(format('The solution is: ~w', .(Sol, [])), ','(nl, ','(format('Solved in ~w msec.', .(T, [])), nl)))))))))).<BR>isPlus(zero, X, X).<BR>isPlus(succ(X), zero, succ(X)).<BR>isPlus(succ(X), succ(Y), succ(succ(Z)))&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(succ(X), pred(Y), Z)&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(pred(X), zero, pred(X)).<BR>isPlus(pred(X), succ(Y), Z)&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(pred(X), pred(Y), pred(pred(Z)))&#160;:-&#160;isPlus(X, Y, Z).<BR>isMinus(X, zero, X).<BR>isMinus(zero, succ(Y), pred(Z))&#160;:-&#160;isMinus(zero, Y, Z).<BR>isMinus(zero, pred(Y), succ(Z))&#160;:-&#160;isMinus(zero, Y, Z).<BR>isMinus(succ(X), succ(Y), Z)&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(succ(X), pred(Y), succ(succ(Z)))&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(pred(X), succ(Y), pred(pred(Z)))&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(pred(X), pred(Y), Z)&#160;:-&#160;isMinus(X, Y, Z).<BR>isTimes(X, zero, zero).<BR>isTimes(zero, succ(Y), zero).<BR>isTimes(zero, pred(Y), zero).<BR>isTimes(succ(X), succ(Y), Z)&#160;:-&#160;','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).<BR>isTimes(succ(X), pred(Y), Z)&#160;:-&#160;','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).<BR>isTimes(pred(X), succ(Y), Z)&#160;:-&#160;','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).<BR>isTimes(pred(X), pred(Y), Z)&#160;:-&#160;','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).<BR>isDiv(zero, succ(Y), zero).<BR>isDiv(zero, pred(Y), zero).<BR>isDiv(succ(X), succ(Y), zero)&#160;:-&#160;isMinus(succ(X), succ(Y), pred(Z)).<BR>isDiv(succ(X), succ(Y), succ(Z))&#160;:-&#160;','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).<BR>isDiv(succ(X), pred(Y), Z)&#160;:-&#160;','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).<BR>isDiv(pred(X), pred(Y), zero)&#160;:-&#160;isMinus(pred(X), pred(Y), succ(Z)).<BR>isDiv(pred(X), pred(Y), succ(Z))&#160;:-&#160;','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).<BR>isDiv(pred(X), succ(Y), Z)&#160;:-&#160;','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).<BR>isModulo(X, Y, Z)&#160;:-&#160;','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).<BR>=(X, X).<BR>isGreater(succ(X), zero).<BR>isGreater(succ(X), pred(Y)).<BR>isGreater(succ(X), succ(Y))&#160;:-&#160;isGreater(X, Y).<BR>isGreater(zero, pred(Y)).<BR>isGreater(pred(X), pred(Y))&#160;:-&#160;isGreater(X, Y).<BR><BR>Queries:<BR><BR>shanoi(g,g,g,g,a).<BR><BR>Added facts for all undefined predicates.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 <B>Prolog</B></pre><pre>          &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>shanoi(succ(zero), A, B, C, .(mv(A, C), [])).<BR>shanoi(N, A, B, C, M)&#160;:-&#160;','(=(X, N), ','(=(X1, succ(zero)), ','(isGreater(X, X1), ','(isMinus(N, succ(zero), U), ','(=(N1, U), ','(shanoi(N1, A, C, B, M1), ','(shanoi(N1, B, A, C, M2), ','(append(M1, .(mv(A, C), []), T), append(T, M2, M))))))))).<BR>append([], L, L).<BR>append(.(H, L), L1, .(H, R))&#160;:-&#160;append(L, L1, R).<BR>demo&#160;:-&#160;','(=(Size, succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(zero)))))))))))))))), ','(format('Towers of Hanoi for ~w elements', .(Size, [])), ','(time(X, A), ','(nl, ','(nl, ','(shanoi(Size, a, b, c, Sol), ','(time(T, Sol), ','(format('The solution is: ~w', .(Sol, [])), ','(nl, ','(format('Solved in ~w msec.', .(T, [])), nl)))))))))).<BR>isPlus(zero, X, X).<BR>isPlus(succ(X), zero, succ(X)).<BR>isPlus(succ(X), succ(Y), succ(succ(Z)))&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(succ(X), pred(Y), Z)&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(pred(X), zero, pred(X)).<BR>isPlus(pred(X), succ(Y), Z)&#160;:-&#160;isPlus(X, Y, Z).<BR>isPlus(pred(X), pred(Y), pred(pred(Z)))&#160;:-&#160;isPlus(X, Y, Z).<BR>isMinus(X, zero, X).<BR>isMinus(zero, succ(Y), pred(Z))&#160;:-&#160;isMinus(zero, Y, Z).<BR>isMinus(zero, pred(Y), succ(Z))&#160;:-&#160;isMinus(zero, Y, Z).<BR>isMinus(succ(X), succ(Y), Z)&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(succ(X), pred(Y), succ(succ(Z)))&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(pred(X), succ(Y), pred(pred(Z)))&#160;:-&#160;isMinus(X, Y, Z).<BR>isMinus(pred(X), pred(Y), Z)&#160;:-&#160;isMinus(X, Y, Z).<BR>isTimes(X, zero, zero).<BR>isTimes(zero, succ(Y), zero).<BR>isTimes(zero, pred(Y), zero).<BR>isTimes(succ(X), succ(Y), Z)&#160;:-&#160;','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).<BR>isTimes(succ(X), pred(Y), Z)&#160;:-&#160;','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).<BR>isTimes(pred(X), succ(Y), Z)&#160;:-&#160;','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).<BR>isTimes(pred(X), pred(Y), Z)&#160;:-&#160;','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).<BR>isDiv(zero, succ(Y), zero).<BR>isDiv(zero, pred(Y), zero).<BR>isDiv(succ(X), succ(Y), zero)&#160;:-&#160;isMinus(succ(X), succ(Y), pred(Z)).<BR>isDiv(succ(X), succ(Y), succ(Z))&#160;:-&#160;','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).<BR>isDiv(succ(X), pred(Y), Z)&#160;:-&#160;','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).<BR>isDiv(pred(X), pred(Y), zero)&#160;:-&#160;isMinus(pred(X), pred(Y), succ(Z)).<BR>isDiv(pred(X), pred(Y), succ(Z))&#160;:-&#160;','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).<BR>isDiv(pred(X), succ(Y), Z)&#160;:-&#160;','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).<BR>isModulo(X, Y, Z)&#160;:-&#160;','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).<BR>=(X, X).<BR>isGreater(succ(X), zero).<BR>isGreater(succ(X), pred(Y)).<BR>isGreater(succ(X), succ(Y))&#160;:-&#160;isGreater(X, Y).<BR>isGreater(zero, pred(Y)).<BR>isGreater(pred(X), pred(Y))&#160;:-&#160;isGreater(X, Y).<BR>format(X0, X1).<BR>time(X0, X1).<BR>nl.<BR><BR>Queries:<BR><BR>shanoi(g,g,g,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>shanoi_in</font>: (b,b,b,b,f)
<BR><FONT COLOR=#0000cc>isGreater_in</font>: (b,b)
<BR><FONT COLOR=#0000cc>isMinus_in</font>: (b,b,f)
<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>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT 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>shanoi_in_gggga</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT 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>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U28_gga</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>U28_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U29_gga</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>U29_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U30_gga</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>U30_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U31_gga</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>U31_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</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>.</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>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>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</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>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT 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>shanoi_in_gggga</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT 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>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U28_gga</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>U28_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U29_gga</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>U29_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U30_gga</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>U30_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U31_gga</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>U31_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</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>.</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>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>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)
<BR><FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</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>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U8_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT 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>shanoi_in_gggga</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT 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>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U28_gga</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>U28_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U29_gga</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>U29_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U30_gga</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>U30_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U31_gga</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>U31_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</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>.</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>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>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U29_GGA</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>U29_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U27_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_GGA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_GGGGA</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U26_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_GGA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U55_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U31_GGA</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>U31_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U28_GGA</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>U28_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT 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>SHANOI_IN_GGGGA</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>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>U1_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_GGGGA</font>(<FONT COLOR=#cc0000>x6</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>U56_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISMINUS_IN_GGA</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>U30_GGA</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>U30_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<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 UndefinedPredicateHandlerProof</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>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>=_IN_AG</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)
<BR><FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</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>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U8_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT 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>shanoi_in_gggga</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT 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>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U28_gga</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>U28_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U29_gga</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>U29_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U30_gga</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>U30_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U31_gga</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>U31_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</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>.</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>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>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U29_GGA</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>U29_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U27_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_GGA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_GGGGA</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U26_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_GGA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U55_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U31_GGA</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>U31_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U28_GGA</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>U28_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT 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>SHANOI_IN_GGGGA</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>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>U1_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U9_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_GGGGA</font>(<FONT COLOR=#cc0000>x6</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>U56_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISMINUS_IN_GGA</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>U30_GGA</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>U30_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 5 SCCs with 19 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 <B>PiDP</B></pre><pre>                        &#8627 UsableRulesProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT 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>shanoi_in_gggga</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT 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>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U28_gga</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>U28_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U29_gga</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>U29_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U30_gga</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>U30_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U31_gga</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>U31_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</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>.</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>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>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x6</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 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 <B>PiDP</B></pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</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 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 PiDP</pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 QDPSizeChangeProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</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>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</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 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 <B>PiDP</B></pre><pre>                        &#8627 UsableRulesProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</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>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT 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>shanoi_in_gggga</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT 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>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U28_gga</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>U28_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U29_gga</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>U29_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U30_gga</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>U30_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U31_gga</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>U31_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</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>.</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>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>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISMINUS_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 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 <B>PiDP</B></pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISMINUS_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 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 PiDP</pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 QDPSizeChangeProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</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>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 >= 1, 2 > 2<P></LI>
<LI><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>zero</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 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 <B>PiDP</B></pre><pre>                        &#8627 UsableRulesProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISMINUS_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>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT 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>shanoi_in_gggga</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT 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>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U28_gga</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>U28_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U29_gga</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>U29_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U30_gga</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>U30_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U31_gga</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>U31_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</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>.</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>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>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISMINUS_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 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 <B>PiDP</B></pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISMINUS_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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</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>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISMINUS_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 UndefinedPredicateHandlerProof</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><pre>                      &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</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>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</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>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISMINUS_IN_GGA</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>ISMINUS_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>ISMINUS_IN_GGA</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>ISMINUS_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>
<LI><FONT COLOR=#0000cc>ISMINUS_IN_GGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISMINUS_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>
<LI><FONT COLOR=#0000cc>ISMINUS_IN_GGA</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>ISMINUS_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>
<LI><FONT COLOR=#0000cc>ISMINUS_IN_GGA</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>ISMINUS_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 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 <B>PiDP</B></pre><pre>                        &#8627 UsableRulesProof</pre><pre>                      &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT 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>shanoi_in_gggga</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT 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>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U28_gga</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>U28_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U29_gga</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>U29_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U30_gga</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>U30_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U31_gga</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>U31_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</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>.</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>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>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 <B>PiDP</B></pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                      &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Pi is empty.<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 UsableRulesProof</pre><pre>                          &#8627 PiDP</pre><pre>                            &#8627 PiDPToQDPProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 QDPSizeChangeProof</pre><pre>                      &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI>
<LI><FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>ISGREATER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 <B>PiDP</B></pre><pre>                        &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT 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>shanoi_in_gggga</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>succ</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT 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>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isGreater_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>isGreater_out_gg</font><BR>
<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U28_gga</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>U28_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U29_gga</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>U29_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U30_gga</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>U30_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U31_gga</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>U31_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U8_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</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>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</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>.</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>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>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT 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>SHANOI_IN_GGGGA</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>U1_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U1_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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>U2_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT 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_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<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 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>)) at position [3] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) at position [5] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</font>)) at position [4] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 <B>QDP</B></pre><pre>                                        &#8627 Instantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 <B>QDP</B></pre><pre>                                            &#8627 Instantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 <B>QDP</B></pre><pre>                                                &#8627 Instantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z4</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 <B>QDP</B></pre><pre>                                                    &#8627 Narrowing</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) at position [3] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 <B>QDP</B></pre><pre>                                                        &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>))) at position [3,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 <B>QDP</B></pre><pre>                                                            &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>))) at position [3,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 <B>QDP</B></pre><pre>                                                                &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>zero</font>))) at position [3,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 <B>QDP</B></pre><pre>                                                                    &#8627 Narrowing</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)) at position [4] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 <B>QDP</B></pre><pre>                                                                        &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))) at position [3] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 <B>QDP</B></pre><pre>                                                                            &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))) at position [3] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>))))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 <B>QDP</B></pre><pre>                                                                                &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>))) at position [3] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 <B>QDP</B></pre><pre>                                                                                    &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>))) at position [4,4] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 <B>QDP</B></pre><pre>                                                                                        &#8627 Narrowing</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) at position [4] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</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>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>x0</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 <B>QDP</B></pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>, <FONT COLOR=#cc0000>z3</font>, <FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 <B>QDP</B></pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 <B>QDP</B></pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 <B>QDP</B></pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 <B>QDP</B></pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>x0</font>))) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 QDP</pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                                  &#8627 <B>QDP</B></pre><pre>                                                                                                                    &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>U</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 QDP</pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                                      &#8627 <B>QDP</B></pre><pre>                                                                                                                        &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>x0</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 QDP</pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                                          &#8627 <B>QDP</B></pre><pre>                                                                                                                            &#8627 ForwardInstantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By forward instantiating [14] the rule <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#cc0000>y1</font>, <FONT COLOR=#cc0000>y2</font>, <FONT COLOR=#cc0000>y3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>zero</font>))) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 QDP</pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                                          &#8627 QDP</pre><pre>                                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                                              &#8627 <B>QDP</B></pre><pre>                                                                                                                                &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>zero</font>))) at position [4,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 QDP</pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                                          &#8627 QDP</pre><pre>                                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                                              &#8627 QDP</pre><pre>                                                                                                                                &#8627 Rewriting</pre><pre>                                                                                                                                  &#8627 <B>QDP</B></pre><pre>                                                                                                                                    &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>zero</font>))) at position [4,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 QDP</pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                                          &#8627 QDP</pre><pre>                                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                                              &#8627 QDP</pre><pre>                                                                                                                                &#8627 Rewriting</pre><pre>                                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                                    &#8627 Rewriting</pre><pre>                                                                                                                                      &#8627 <B>QDP</B></pre><pre>                                                                                                                                        &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>)) at position [4] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 QDP</pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                                          &#8627 QDP</pre><pre>                                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                                              &#8627 QDP</pre><pre>                                                                                                                                &#8627 Rewriting</pre><pre>                                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                                    &#8627 Rewriting</pre><pre>                                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                                        &#8627 Rewriting</pre><pre>                                                                                                                                          &#8627 <B>QDP</B></pre><pre>                                                                                                                                            &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>)
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>)) at position [4] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 QDP</pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                                          &#8627 QDP</pre><pre>                                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                                              &#8627 QDP</pre><pre>                                                                                                                                &#8627 Rewriting</pre><pre>                                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                                    &#8627 Rewriting</pre><pre>                                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                                        &#8627 Rewriting</pre><pre>                                                                                                                                          &#8627 QDP</pre><pre>                                                                                                                                            &#8627 Rewriting</pre><pre>                                                                                                                                              &#8627 <B>QDP</B></pre><pre>                                                                                                                                                &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</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>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>)</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<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>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</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>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U26_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U27_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U28_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U29_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>, x<SUB>6</SUB>)</B>) = x<SUB>5</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>, x<SUB>6</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U30_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U31_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U55_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U56_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_gggga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>[]</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>append_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>append_out_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isGreater_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isGreater_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isMinus_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isMinus_out_gga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>mv</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>pred</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>shanoi_in_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>shanoi_out_gggga</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>) = 1 + x<SUB>1</SUB><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:
none<BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 UndefinedPredicateHandlerProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 PrologToPiTRSProof</pre><pre>            &#8627 PiTRS</pre><pre>              &#8627 DependencyPairsProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 DependencyGraphProof</pre><pre>                    &#8627 AND</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                      &#8627 PiDP</pre><pre>                        &#8627 PiDPToQDPProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Rewriting</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 Rewriting</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 Instantiation</pre><pre>                                          &#8627 QDP</pre><pre>                                            &#8627 Instantiation</pre><pre>                                              &#8627 QDP</pre><pre>                                                &#8627 Instantiation</pre><pre>                                                  &#8627 QDP</pre><pre>                                                    &#8627 Narrowing</pre><pre>                                                      &#8627 QDP</pre><pre>                                                        &#8627 Rewriting</pre><pre>                                                          &#8627 QDP</pre><pre>                                                            &#8627 Rewriting</pre><pre>                                                              &#8627 QDP</pre><pre>                                                                &#8627 Rewriting</pre><pre>                                                                  &#8627 QDP</pre><pre>                                                                    &#8627 Narrowing</pre><pre>                                                                      &#8627 QDP</pre><pre>                                                                        &#8627 Rewriting</pre><pre>                                                                          &#8627 QDP</pre><pre>                                                                            &#8627 Rewriting</pre><pre>                                                                              &#8627 QDP</pre><pre>                                                                                &#8627 Rewriting</pre><pre>                                                                                  &#8627 QDP</pre><pre>                                                                                    &#8627 Rewriting</pre><pre>                                                                                      &#8627 QDP</pre><pre>                                                                                        &#8627 Narrowing</pre><pre>                                                                                          &#8627 QDP</pre><pre>                                                                                            &#8627 DependencyGraphProof</pre><pre>                                                                                              &#8627 QDP</pre><pre>                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                  &#8627 QDP</pre><pre>                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                      &#8627 QDP</pre><pre>                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                          &#8627 QDP</pre><pre>                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                              &#8627 QDP</pre><pre>                                                                                                                &#8627 ForwardInstantiation</pre><pre>                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                    &#8627 ForwardInstantiation</pre><pre>                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                        &#8627 ForwardInstantiation</pre><pre>                                                                                                                          &#8627 QDP</pre><pre>                                                                                                                            &#8627 ForwardInstantiation</pre><pre>                                                                                                                              &#8627 QDP</pre><pre>                                                                                                                                &#8627 Rewriting</pre><pre>                                                                                                                                  &#8627 QDP</pre><pre>                                                                                                                                    &#8627 Rewriting</pre><pre>                                                                                                                                      &#8627 QDP</pre><pre>                                                                                                                                        &#8627 Rewriting</pre><pre>                                                                                                                                          &#8627 QDP</pre><pre>                                                                                                                                            &#8627 Rewriting</pre><pre>                                                                                                                                              &#8627 QDP</pre><pre>                                                                                                                                                &#8627 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_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>)
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))) &#8594; <FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>), <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_3</font>))))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>)), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_GGGGA</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x4</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>y_0</font>), <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N</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>U1_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>isGreater_out_gg</font>
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isGreater_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isMinus_in_gga</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>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isMinus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U8_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U9_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</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>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</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>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)
<BR><FONT COLOR=#0000cc>isGreater_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U56_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U55_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</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>isMinus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U31_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U30_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U29_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U28_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U27_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U26_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</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>U5_gggga</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_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U7_gggga</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>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>U8_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gggga</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 10 less nodes.<BR><BR></body>

