YES
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<title>Left Termination proof of ../tpdb/LP/BCGGV05/map_color.pl</title>
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<BR><B>Left Termination</B> of the query pattern
map_in_1(a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>color_map([], X).<BR>color_map(.(Region, Regions), Colors)&#160;:-&#160;','(color_region(Region, Colors), color_map(Regions, Colors)).<BR>color_region(region(X, Color, Neighbors), Colors)&#160;:-&#160;','(select(Color, Colors, Colors1), subset(Neighbors, Colors1)).<BR>select(X, .(X, Xs), Xs).<BR>select(X, .(Y, Xs), .(Y, Zs))&#160;:-&#160;select(X, Xs, Zs).<BR>subset([], X).<BR>subset(.(X, Xs), Ys)&#160;:-&#160;','(member(X, Ys), subset(Xs, Ys)).<BR>member(X, .(X, X1)).<BR>member(X, .(X1, Xs))&#160;:-&#160;member(X, Xs).<BR>map(.(region(belize, Belize, .(Guatemala, [])), .(region(guatemala, Guatemala, .(Belize, .(El_Salvador, .(Honduras, [])))), .(region(el_salvador, El_Salvador, .(Guatemala, .(Honduras, []))), .(region(honduras, Honduras, .(Guatemala, .(El_Salvador, .(Nicaragua, [])))), .(region(nicaragua, Nicaragua, .(Honduras, .(Costa_rica, []))), .(region(costa_rica, Costa_rica, .(Nicaragua, .(Panama, []))), .(region(panama, Panama, .(Costa_rica, [])), [])))))))).<BR>query&#160;:-&#160;','(map(X), color_map(X, .(red, .(blue, .(green, []))))).<BR><BR>Queries:<BR><BR>map(a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR>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>map_in_a</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>belize</font>, <FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>guatemala</font>, <FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>el_salvador</font>, <FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>honduras</font>, <FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>nicaragua</font>, <FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>costa_rica</font>, <FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>panama</font>, <FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>[]</font>)))))))) &#8594; <FONT COLOR=#0000cc>map_out_a</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>belize</font>, <FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>guatemala</font>, <FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>el_salvador</font>, <FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>honduras</font>, <FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>nicaragua</font>, <FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>costa_rica</font>, <FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>panama</font>, <FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>[]</font>))))))))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>map_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>map_in_a</font><BR>
<FONT COLOR=#0000cc>map_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>map_out_a</font><BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>map_in_a</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>belize</font>, <FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>guatemala</font>, <FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>el_salvador</font>, <FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>honduras</font>, <FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>nicaragua</font>, <FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>costa_rica</font>, <FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>panama</font>, <FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>[]</font>)))))))) &#8594; <FONT COLOR=#0000cc>map_out_a</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>belize</font>, <FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>guatemala</font>, <FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>el_salvador</font>, <FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>honduras</font>, <FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>nicaragua</font>, <FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>costa_rica</font>, <FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>panama</font>, <FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>[]</font>))))))))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>map_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>map_in_a</font><BR>
<FONT COLOR=#0000cc>map_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>map_out_a</font><BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>P is empty.<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>map_in_a</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>belize</font>, <FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>guatemala</font>, <FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>el_salvador</font>, <FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>honduras</font>, <FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>nicaragua</font>, <FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>costa_rica</font>, <FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>panama</font>, <FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>[]</font>)))))))) &#8594; <FONT COLOR=#0000cc>map_out_a</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>belize</font>, <FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>guatemala</font>, <FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>el_salvador</font>, <FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>honduras</font>, <FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>nicaragua</font>, <FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>costa_rica</font>, <FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>panama</font>, <FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>[]</font>))))))))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>map_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>map_in_a</font><BR>
<FONT COLOR=#0000cc>map_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>map_out_a</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 PisEmptyProof</pre><BR>Pi DP problem:<BR>P is empty.<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>map_in_a</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>belize</font>, <FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>guatemala</font>, <FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>el_salvador</font>, <FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>honduras</font>, <FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>nicaragua</font>, <FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>costa_rica</font>, <FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>panama</font>, <FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>[]</font>)))))))) &#8594; <FONT COLOR=#0000cc>map_out_a</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>belize</font>, <FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>guatemala</font>, <FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Belize</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>el_salvador</font>, <FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>honduras</font>, <FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Guatemala</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>El_Salvador</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>[]</font>)))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>nicaragua</font>, <FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Honduras</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>costa_rica</font>, <FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Nicaragua</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>[]</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>region</font>(<FONT COLOR=#0000cc>panama</font>, <FONT COLOR=#cc0000>Panama</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Costa_rica</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>[]</font>))))))))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>map_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>map_in_a</font><BR>
<FONT COLOR=#0000cc>map_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>map_out_a</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The TRS P is empty. Hence, there is no (P,R,Pi) chain.<BR><BR></body>

