MAYBE
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<title>Left Termination proof of ../tpdb/LP/SGST06/mapcolor.pl</title>
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<BR><B>Left Termination</B> of the query pattern
color_map_in_2(a, g)
w.r.t. the given <I>Prolog program</I> could not be shown:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>color_map(.(Region, Regions), Colors)&#160;:-&#160;','(color_region(Region, Colors), color_map(Regions, Colors)).<BR>color_map([], Colors).<BR>color_region(region(Name, Color, Neighbors), Colors)&#160;:-&#160;','(select(Color, Colors, Colors1), members(Neighbors, Colors1)).<BR>select(X, .(X, Xs), Xs).<BR>select(X, .(Y, Ys), .(Y, Zs))&#160;:-&#160;select(X, Ys, Zs).<BR>members(.(X, Xs), Ys)&#160;:-&#160;','(member(X, Ys), members(Xs, Ys)).<BR>members([], Ys).<BR>member(X, .(X, X1)).<BR>member(X, .(X1, Xs))&#160;:-&#160;member(X, Xs).<BR><BR>Queries:<BR><BR>color_map(a,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>color_map_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>color_region_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>select_in</font>: (f,b,f)
<BR><FONT COLOR=#0000cc>members_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>member_in</font>: (f,b)
<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>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</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><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</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>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)
<BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_AG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</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><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)
<BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_AG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 4 SCCs with 10 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><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_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 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</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>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_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 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>Xs</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_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBERS_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><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBERS_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 Narrowing</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member_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_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 Instantiation</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member_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_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Instantiation</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 NonTerminationProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member_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_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by narrowing to the left:<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR><BR>s = <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) evaluates to  t =<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>))<BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR><B><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>))</B> &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>))<BR>with rule <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0'</font>, <FONT COLOR=#cc0000>x1'</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0'</font>, <FONT COLOR=#cc0000>x1'</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0'</font>)) at position [] and matcher [<FONT COLOR=#cc0000>x1'</font> / <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x0'</font> / <FONT COLOR=#cc0000>x0</font>]<BR><BR><B><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>))</B> &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>))<BR>with rule <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))<BR><BR>Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence<BR><BR><BR>All these steps are and every following step will be a correct step w.r.t to Q.<BR><BR><BR><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 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</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>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>Ys</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>SELECT_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>Ys</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 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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 Narrowing</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Colors</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Colors</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x0</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x0</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 Narrowing</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 UsableRulesProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x0</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] 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 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 UsableRulesProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 QReductionProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x0</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x0</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 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 UsableRulesProof</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 QReductionProof</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x0</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<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>color_map_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>color_region_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>select_in</font>: (f,b,f)
<BR><FONT COLOR=#0000cc>members_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>member_in</font>: (f,b)
<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>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 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>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)
<BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</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>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)
<BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_REGION_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 4 SCCs with 10 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</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>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_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 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>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</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>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBER_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 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>MEMBER_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>Xs</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_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>MEMBER_IN_AG</font>(<FONT COLOR=#cc0000>Xs</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 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>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBERS_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 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>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MEMBERS_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 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 Narrowing</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>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 Instantiation</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>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Instantiation</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 NonTerminationProof</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>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by narrowing to the left:<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))
<BR><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)))
<BR><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR><BR>s = <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1'</font>)) evaluates to  t =<FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1'</font>))<BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR><B><FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1'</font>))</B> &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1'</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1'</font>)))<BR>with rule <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0'</font>, <FONT COLOR=#cc0000>x1''</font>)) &#8594; <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0'</font>, <FONT COLOR=#cc0000>x1''</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0'</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0'</font>, <FONT COLOR=#cc0000>x1''</font>))) at position [] and matcher [<FONT COLOR=#cc0000>x1''</font> / <FONT COLOR=#cc0000>x1'</font>, <FONT COLOR=#cc0000>x0'</font> / <FONT COLOR=#cc0000>x0</font>]<BR><BR><B><FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1'</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1'</font>)))</B> &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1'</font>))<BR>with rule <FONT COLOR=#0000cc>U6_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>), <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))) &#8594; <FONT COLOR=#0000cc>MEMBERS_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>z0</font>, <FONT COLOR=#cc0000>z1</font>))<BR><BR>Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence<BR><BR><BR>All these steps are and every following step will be a correct step w.r.t to Q.<BR><BR><BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</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>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</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>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</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>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</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>SELECT_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>Ys</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>SELECT_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>SELECT_IN_AGA</font>(<FONT COLOR=#cc0000>Ys</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 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>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_map_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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 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>COLOR_MAP_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Regions</font>, <FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Name</font>, <FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_in_aga</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>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_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>U6_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_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>U8_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_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>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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 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><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#cc0000>Region</font>, <FONT COLOR=#cc0000>Colors</font>)) &#8594; <FONT COLOR=#0000cc>COLOR_MAP_IN_AG</font>(<FONT COLOR=#cc0000>Colors</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>Colors</font>) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Colors</font>))
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Colors1</font>))
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Colors</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Neighbors</font>, <FONT COLOR=#cc0000>Colors1</font>)) &#8594; <FONT COLOR=#0000cc>color_region_out_ag</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#cc0000>Color</font>, <FONT COLOR=#cc0000>Neighbors</font>), <FONT COLOR=#cc0000>Colors</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>select_out_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>members_out_ag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>member_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>color_region_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>select_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>members_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>member_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U8_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR><BR></body>

