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<title>Left Termination proof of ../tpdb/LP/terminweb/old-terminweb/queens.pl</title>
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<BR><B>Left Termination</B> of the query pattern
queens_in_2(g, a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><BR>Clauses:<BR><BR>queens(X, Y)&#160;:-&#160;','(perm(X, Y), safe(Y)).<BR>perm([], []).<BR>perm(.(X, Y), .(V, Res))&#160;:-&#160;','(delete(V, .(X, Y), Rest), perm(Rest, Res)).<BR>delete(X, .(X, Y), Y).<BR>delete(X, .(F, T), .(F, R))&#160;:-&#160;delete(X, T, R).<BR>safe([]).<BR>safe(.(X, Y))&#160;:-&#160;','(noattack(X, Y, 1), safe(Y)).<BR>noattack(X, [], N).<BR>noattack(X, .(F, T), N)&#160;:-&#160;','(=\=(X, F), ','(=\=(X, +(F, N)), ','(=\=(F, +(X, N)), ','(is(N1, +(N, 1)), noattack(X, T, N1))))).<BR><BR>Queries:<BR><BR>queens(g,a).<BR><BR>Added definitions of predefined predicates.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 <B>Prolog</B></pre><pre>      &#8627 CutEliminatorProof</pre><BR>Clauses:<BR><BR>queens(X, Y)&#160;:-&#160;','(perm(X, Y), safe(Y)).<BR>perm([], []).<BR>perm(.(X, Y), .(V, Res))&#160;:-&#160;','(delete(V, .(X, Y), Rest), perm(Rest, Res)).<BR>delete(X, .(X, Y), Y).<BR>delete(X, .(F, T), .(F, R))&#160;:-&#160;delete(X, T, R).<BR>safe([]).<BR>safe(.(X, Y))&#160;:-&#160;','(noattack(X, Y, succ(zero)), safe(Y)).<BR>noattack(X, [], N).<BR>noattack(X, .(F, T), N)&#160;:-&#160;','(=\=(X, F), ','(=\=(X, +(F, N)), ','(=\=(F, +(X, N)), ','(','(isPlus(N, succ(zero), U), =(N1, U)), noattack(X, T, N1))))).<BR>nonequal(X, Y)&#160;:-&#160;','(=(X, Y), ','(!, fail)).<BR>nonequal(X, Y)&#160;:-&#160;true.<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>fail&#160;:-&#160;fail(b).<BR>fail(a).<BR>true.<BR>=(X, X).<BR><BR>Queries:<BR><BR>queens(g,a).<BR><BR>Eliminated all cuts by simply ignoring them.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 <B>Prolog</B></pre><pre>          &#8627 UndefinedPredicateHandlerProof</pre><BR>Clauses:<BR><BR>queens(X, Y)&#160;:-&#160;','(perm(X, Y), safe(Y)).<BR>perm([], []).<BR>perm(.(X, Y), .(V, Res))&#160;:-&#160;','(delete(V, .(X, Y), Rest), perm(Rest, Res)).<BR>delete(X, .(X, Y), Y).<BR>delete(X, .(F, T), .(F, R))&#160;:-&#160;delete(X, T, R).<BR>safe([]).<BR>safe(.(X, Y))&#160;:-&#160;','(noattack(X, Y, succ(zero)), safe(Y)).<BR>noattack(X, [], N).<BR>noattack(X, .(F, T), N)&#160;:-&#160;','(=\=(X, F), ','(=\=(X, +(F, N)), ','(=\=(F, +(X, N)), ','(','(isPlus(N, succ(zero), U), =(N1, U)), noattack(X, T, N1))))).<BR>nonequal(X, Y)&#160;:-&#160;','(=(X, Y), fail).<BR>nonequal(X, Y)&#160;:-&#160;true.<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>fail&#160;:-&#160;fail(b).<BR>fail(a).<BR>true.<BR>=(X, X).<BR><BR>Queries:<BR><BR>queens(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 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 UndefinedPredicateHandlerProof</pre><pre>            &#8627 <B>Prolog</B></pre><pre>              &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>queens(X, Y)&#160;:-&#160;','(perm(X, Y), safe(Y)).<BR>perm([], []).<BR>perm(.(X, Y), .(V, Res))&#160;:-&#160;','(delete(V, .(X, Y), Rest), perm(Rest, Res)).<BR>delete(X, .(X, Y), Y).<BR>delete(X, .(F, T), .(F, R))&#160;:-&#160;delete(X, T, R).<BR>safe([]).<BR>safe(.(X, Y))&#160;:-&#160;','(noattack(X, Y, succ(zero)), safe(Y)).<BR>noattack(X, [], N).<BR>noattack(X, .(F, T), N)&#160;:-&#160;','(=\=(X, F), ','(=\=(X, +(F, N)), ','(=\=(F, +(X, N)), ','(isPlus(N, succ(zero), U), ','(=(N1, U), noattack(X, T, N1)))))).<BR>nonequal(X, Y)&#160;:-&#160;','(=(X, Y), fail).<BR>nonequal(X, Y)&#160;:-&#160;true.<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>fail&#160;:-&#160;fail(b).<BR>fail(a).<BR>true.<BR>=(X, X).<BR>=\=(X0, X1).<BR><BR>Queries:<BR><BR>queens(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>queens_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>perm_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>delete_in</font>: (f,b,f)
<BR><FONT COLOR=#0000cc>safe_in</font>: (b)
<BR><FONT COLOR=#0000cc>noattack_in</font>: (b,b,b)
<BR><FONT COLOR=#0000cc>isPlus_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>queens_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_out_g</font><BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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>queens_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_out_g</font><BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>QUEENS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>QUEENS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)
<BR><FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))
<BR><FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)
<BR><FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_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>ISPLUS_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISPLUS_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISPLUS_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)
<BR><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_out_g</font><BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U12_GGG</font>(<FONT 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>U12_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_GGG</font>(<FONT 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_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>QUEENS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>QUEENS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U17_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>U17_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DELETE_IN_AGA</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>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_GGG</font>(<FONT 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>U11_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U13_GGG</font>(<FONT 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>U13_GGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGG</font>(<FONT 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>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U9_GGG</font>(<FONT 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>U9_GGG</font>(<FONT 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>U18_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>U18_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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>QUEENS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>QUEENS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)
<BR><FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))
<BR><FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)
<BR><FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_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>ISPLUS_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISPLUS_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ISPLUS_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)
<BR><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_out_g</font><BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U12_GGG</font>(<FONT 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>U12_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_GGG</font>(<FONT 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_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>QUEENS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>QUEENS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U17_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>U17_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DELETE_IN_AGA</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>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_GGG</font>(<FONT 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>U11_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U13_GGG</font>(<FONT 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>U13_GGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGG</font>(<FONT 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>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U9_GGG</font>(<FONT 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>U9_GGG</font>(<FONT 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>U18_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>U18_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 5 SCCs with 19 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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>ISPLUS_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_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>queens_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_out_g</font><BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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>ISPLUS_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>ISPLUS_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>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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>ISPLUS_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>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>ISPLUS_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>ISPLUS_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>ISPLUS_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>ISPLUS_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>ISPLUS_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>ISPLUS_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>ISPLUS_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>ISPLUS_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>ISPLUS_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>ISPLUS_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)
<BR><FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_out_g</font><BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_GGG</font>(<FONT 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>U12_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_GGG</font>(<FONT 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_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U11_GGG</font>(<FONT 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>U11_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGG</font>(<FONT 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>U9_GGG</font>(<FONT 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>U9_GGG</font>(<FONT 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>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)
<BR><FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</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>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>+</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>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</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>U12_GGG</font>(<FONT 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>U12_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_GGG</font>(<FONT 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_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U11_GGG</font>(<FONT 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>U11_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGG</font>(<FONT 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>U9_GGG</font>(<FONT 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>U9_GGG</font>(<FONT 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>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)
<BR><FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_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>U17_gga</font>(<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isPlus_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>U19_gga</font>(<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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>U18_gga</font>(<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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>U20_gga</font>(<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))<BR>The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4<P></LI>
<LI><FONT COLOR=#0000cc>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)<BR>The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3<P></LI>
<LI><FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))<BR>The graph contains the following edges 1 >= 1, 2 >= 2<P></LI>
<LI><FONT COLOR=#0000cc>U11_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U12_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))<BR>The graph contains the following edges 1 >= 1, 2 >= 2<P></LI>
<LI><FONT COLOR=#0000cc>U9_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U10_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))<BR>The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3<P></LI>
<LI><FONT COLOR=#0000cc>NOATTACK_IN_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_GGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))<BR>The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>queens_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_out_g</font><BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</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>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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>SAFE_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>) &#8594; <FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_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>U17_gga</font>(<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isPlus_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>U19_gga</font>(<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</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>U13_ggg</font>(<FONT COLOR=#0000cc>noattack_out_ggg</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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>U18_gga</font>(<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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>U20_gga</font>(<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT 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>=\=_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U10_ggg</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>U11_ggg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_ggg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>) &#8594; <FONT COLOR=#0000cc>SAFE_IN_G</font>(<FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 >= 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_out_g</font><BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</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>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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>DELETE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_AGA</font>(<FONT COLOR=#cc0000>T</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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 UsableRulesProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>)))
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>))
<BR><FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>)) &#8594; <FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U9_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>)))
<BR><FONT COLOR=#0000cc>U10_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>N</font>))) &#8594; <FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>isPlus_in_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U20_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>pred</font>(<FONT COLOR=#0000cc>pred</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U19_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U18_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U17_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>isPlus_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=#0000cc>succ</font>(<FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U11_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>isPlus_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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</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>U12_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>=_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>))
<BR><FONT COLOR=#0000cc>U13_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>succ</font>(<FONT COLOR=#0000cc>zero</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>safe_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>safe_out_g</font><BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>noattack_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noattack_out_ggg</font><BR>
<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>=\=_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>=\=_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_ggg</font>(<FONT 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>U9_ggg</font>(<FONT 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>+</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_ggg</font>(<FONT 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_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U11_ggg</font>(<FONT 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>U11_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>isPlus_out_gga</font>(<FONT COLOR=#cc0000>x3</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>U17_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>U17_gga</font>(<FONT COLOR=#cc0000>x4</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>U18_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>U18_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U19_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>U19_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U20_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>U20_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U12_ggg</font>(<FONT 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>U12_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>U13_ggg</font>(<FONT 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>U13_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>queens_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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 UsableRulesProof</pre><pre>                              &#8627 <B>PiDP</B></pre><pre>                                &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#cc0000>Res</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Res</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Rest</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</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 UsableRulesProof</pre><pre>                              &#8627 PiDP</pre><pre>                                &#8627 PiDPToQDPProof</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 RuleRemovalProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Rest</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
<BR><BR>Strictly oriented rules of the TRS R:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>PERM_IN_GA</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_GA</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_aga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>delete_in_aga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>delete_out_aga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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 UsableRulesProof</pre><pre>                              &#8627 PiDP</pre><pre>                                &#8627 PiDPToQDPProof</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 RuleRemovalProof</pre><pre>                                      &#8627 <B>QDP</B></pre><pre>                                        &#8627 RuleRemovalProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Rest</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
<BR>Strictly oriented dependency pairs:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>Rest</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Rest</font>)</BLOCKQUOTE><BR><BR>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>PERM_IN_GA</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_aga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>delete_in_aga</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>delete_out_aga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2 + x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 CutEliminatorProof</pre><pre>        &#8627 Prolog</pre><pre>          &#8627 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 UsableRulesProof</pre><pre>                              &#8627 PiDP</pre><pre>                                &#8627 PiDPToQDPProof</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 RuleRemovalProof</pre><pre>                                      &#8627 QDP</pre><pre>                                        &#8627 RuleRemovalProof</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>PERM_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#0000cc>.</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>delete_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>F</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>delete_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</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 1 less node.<BR><BR></body>

