YES
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<title>Left Termination proof of ../tpdb/LP/talp/plumer/pl7.6.2c.pl</title>
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<BR><B>Left Termination</B> of the query pattern
reach_in_4(g, g, g, g)
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>reach(X, Y, E, L)&#160;:-&#160;member(.(X, .(Y, [])), E).<BR>reach(X, Z, E, L)&#160;:-&#160;','(member1(.(X, .(Y, [])), E), ','(member(Y, L), ','(delete(Y, L, V1), reach(Y, Z, E, V1)))).<BR>member(H, .(H, L)).<BR>member(X, .(H, L))&#160;:-&#160;member(X, L).<BR>member1(H, .(H, L)).<BR>member1(X, .(H, L))&#160;:-&#160;member1(X, L).<BR>delete(X, .(X, Y), Y).<BR>delete(X, .(H, T1), .(H, T2))&#160;:-&#160;delete(X, T1, T2).<BR><BR>Queries:<BR><BR>reach(g,g,g,g).<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>reach_in</font>: (b,b,b,b)
<BR><FONT COLOR=#0000cc>member_in</font>: (b,b)
<BR><FONT COLOR=#0000cc>member1_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>delete_in</font>: (b,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>reach_in_gggg</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>reach_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</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_gg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>reach_out_gggg</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>reach_out_gggg</font><BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ag</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>U7_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 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>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>reach_in_gggg</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>reach_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</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_gg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>reach_out_gggg</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>reach_out_gggg</font><BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ag</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>U7_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)
<BR><FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)
<BR><FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)
<BR><FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)
<BR><FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>reach_in_gggg</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>reach_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</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_gg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>reach_out_gggg</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>reach_out_gggg</font><BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ag</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>U7_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_AG</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>U7_AG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_GG</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_GG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>REACH_IN_GGGG</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>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 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>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)
<BR><FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)
<BR><FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)
<BR><FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)
<BR><FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>reach_in_gggg</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>reach_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</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_gg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>reach_out_gggg</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>reach_out_gggg</font><BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ag</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>U7_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_AG</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>U7_AG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_GG</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_GG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>REACH_IN_GGGG</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>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 4 SCCs with 9 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><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>reach_in_gggg</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>reach_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</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_gg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>reach_out_gggg</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>reach_out_gggg</font><BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ag</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>U7_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>DELETE_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)<BR>The graph contains the following edges 1 >= 1, 2 > 2<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><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>reach_in_gggg</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>reach_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</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_gg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>reach_out_gggg</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>reach_out_gggg</font><BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ag</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>U7_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>L</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>MEMBER1_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER1_IN_AG</font>(<FONT COLOR=#cc0000>L</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 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>reach_in_gggg</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>reach_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</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_gg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>reach_out_gggg</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>reach_out_gggg</font><BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ag</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>U7_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>R is empty.<BR>Pi is empty.<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</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>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)<BR>The graph contains the following edges 1 >= 1, 2 > 2<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 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_in_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>reach_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>reach_in_gggg</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>reach_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</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_gg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>reach_out_gggg</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>reach_out_gggg</font><BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ag</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>U7_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>REACH_IN_GGGG</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>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</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 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>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>)) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>))
<BR><FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>E</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</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_gg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_ag</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>U7_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>REACH_IN_GGGG</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>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>E</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>)
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#0000cc>member_out_gg</font>) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)
<BR><FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>E</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GGGG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_GGGG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 1 + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_GGGG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_ag</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>[]</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>delete_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>delete_out_gga</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>member1_in_ag</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>member1_out_ag</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>member_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>member_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</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 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 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>V1</font>)) &#8594; <FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>V1</font>)
<BR><FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>REACH_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>E</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>
<BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>H</font>)
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#0000cc>member_out_gg</font>) &#8594; <FONT COLOR=#0000cc>member_out_gg</font>
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>delete_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member1_out_ag</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>delete_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>member1_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.<BR><BR></body>

