YES
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<title>Left Termination proof of ../tpdb/LP/SGST06/cnfequiv.pl</title>
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<BR><B>Left Termination</B> of the query pattern
cnfequiv_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 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>cnfequiv(X, Y)&#160;:-&#160;','(transform(X, Z), cnfequiv(Z, Y)).<BR>cnfequiv(X, X).<BR>transform(n(n(X)), X).<BR>transform(n(a(X, Y)), o(n(X), n(Y))).<BR>transform(n(o(X, Y)), a(n(X), n(Y))).<BR>transform(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))).<BR>transform(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))).<BR>transform(o(X1, Y), o(X2, Y))&#160;:-&#160;transform(X1, X2).<BR>transform(o(X, Y1), o(X, Y2))&#160;:-&#160;transform(Y1, Y2).<BR>transform(a(X1, Y), a(X2, Y))&#160;:-&#160;transform(X1, X2).<BR>transform(a(X, Y1), a(X, Y2))&#160;:-&#160;transform(Y1, Y2).<BR>transform(n(X1), n(X2))&#160;:-&#160;transform(X1, X2).<BR><BR>Queries:<BR><BR>cnfequiv(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>cnfequiv_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>transform_in</font>: (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>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</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>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>cnfequiv_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>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>o</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x3</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>x3</font>)<BR>
<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_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 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>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</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>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>cnfequiv_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>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>o</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x3</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>x3</font>)<BR>
<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_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>CNFEQUIV_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>Z</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>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</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>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>cnfequiv_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>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>o</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x3</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>x3</font>)<BR>
<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSFORM_IN_GA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>x1</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>x3</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>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>CNFEQUIV_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>Z</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>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</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>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>cnfequiv_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>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>o</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x3</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>x3</font>)<BR>
<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSFORM_IN_GA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>x1</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>x3</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>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 2 SCCs with 7 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</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>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>cnfequiv_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>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>o</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x3</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>x3</font>)<BR>
<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSFORM_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 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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSFORM_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 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><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</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>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>X1</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>TRANSFORM_IN_GA</font>(<FONT COLOR=#cc0000>Y1</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <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>CNFEQUIV_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</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>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>cnfequiv_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>cnfequiv_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_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>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>o</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x3</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>x3</font>)<BR>
<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cnfequiv_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CNFEQUIV_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>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>CNFEQUIV_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>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_in_ga</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>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>), <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>o</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>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CNFEQUIV_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>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR>none<BR>Used ordering:  Combined order from the following AFS and order.<BR><FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>CNFEQUIV_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#cc0000>x1</font><BR>
<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>Recursive path order with status [2].<BR>Quasi-Precedence: <BLOCKQUOTE>
[n<SUB><FONT SIZE="-1">1</FONT></SUB>, U7<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">1]</FONT></SUB></FONT></SUB> &gt; [o<SUB><FONT SIZE="-1">2</FONT></SUB>, U4<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB>, U3<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">2]</FONT></SUB></FONT></SUB> &gt; [a<SUB><FONT SIZE="-1">2</FONT></SUB>, U5<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB>, U6<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">2]</FONT></SUB></FONT></SUB> &gt; [CNFEQUIV<SUB><FONT SIZE="-1">IN<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB></FONT></SUB>, transform<SUB><FONT SIZE="-1">out<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">1]</FONT></SUB></FONT></SUB></FONT></SUB> &gt; U1<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB><BR>
</BLOCKQUOTE>
<BR>Status: <BLOCKQUOTE>U5<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB>: multiset<BR>U6<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB>: multiset<BR>a<SUB><FONT SIZE="-1">2</FONT></SUB>: multiset<BR>transform<SUB><FONT SIZE="-1">out<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB></FONT></SUB>: multiset<BR>U4<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB>: multiset<BR>n<SUB><FONT SIZE="-1">1</FONT></SUB>: [1]<BR>U7<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB>: [1]<BR>U3<SUB><FONT SIZE="-1">ga<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB>: multiset<BR>o<SUB><FONT SIZE="-1">2</FONT></SUB>: multiset<BR>CNFEQUIV<SUB><FONT SIZE="-1">IN<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB></FONT></SUB>: multiset<BR>U1<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB>: multiset</BLOCKQUOTE><BR><BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 PisEmptyProof</pre><BR>Q DP problem:<BR>P is empty.<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>), <FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>o</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>a</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>transform_out_ga</font>(<FONT COLOR=#0000cc>n</font>(<FONT COLOR=#cc0000>X2</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transform_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The TRS P is empty. Hence, there is no (P,Q,R) chain.<BR><BR></body>

