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<title>Left Termination proof of ../tpdb/LP/talp/maria/grammar.pl</title>
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<BR><B>Left Termination</B> of the query pattern

w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><BR>Clauses:<BR><BR>goal&#160;:-&#160;','(np(S1, S2, S), verb(S2, S3, S)).<BR>parse(S0, Meaning)&#160;:-&#160;','(np(S0, S1, Meaning), verb(S1, [], Meaning)).<BR>parse(S0, Meaning)&#160;:-&#160;','(np(S0, S1, Meaning), ','(verb(S1, S2, Meaning), =(S2, []))).<BR>np(Si, So, S)&#160;:-&#160;','(det(Si, St, T), ','(noun(St, So, N), comb(T, N, S))).<BR>comb(a, -(N, s), s(s, N, V)).<BR>comb(the, -(N, P), s(P, N, V)).<BR>det(.(a, S), S, a).<BR>det(.(the, S), S, the).<BR>noun(.(book, S), S, -(book, s)).<BR>noun(.(books, S), S, -(book, p)).<BR>noun(.(box, S), S, -(box, s)).<BR>noun(.(boxes, S), S, -(box, p)).<BR>verb(.(falls, So), So, s(s, N, fall)).<BR>verb(.(fall, So), So, s(p, N, fall)).<BR>verb(.(flies, So), So, s(s, N, fly)).<BR>verb(.(fly, So), So, s(p, N, fly)).<BR><BR>Queries:<BR><BR>goal().<BR><BR>Added definitions of predefined predicates.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 <B>Prolog</B></pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>goal&#160;:-&#160;','(np(S1, S2, S), verb(S2, S3, S)).<BR>parse(S0, Meaning)&#160;:-&#160;','(np(S0, S1, Meaning), verb(S1, [], Meaning)).<BR>parse(S0, Meaning)&#160;:-&#160;','(np(S0, S1, Meaning), ','(verb(S1, S2, Meaning), =(S2, []))).<BR>np(Si, So, S)&#160;:-&#160;','(det(Si, St, T), ','(noun(St, So, N), comb(T, N, S))).<BR>comb(a, -(N, s), s(s, N, V)).<BR>comb(the, -(N, P), s(P, N, V)).<BR>det(.(a, S), S, a).<BR>det(.(the, S), S, the).<BR>noun(.(book, S), S, -(book, s)).<BR>noun(.(books, S), S, -(book, p)).<BR>noun(.(box, S), S, -(box, s)).<BR>noun(.(boxes, S), S, -(box, p)).<BR>verb(.(falls, So), So, s(s, N, fall)).<BR>verb(.(fall, So), So, s(p, N, fall)).<BR>verb(.(flies, So), So, s(s, N, fly)).<BR>verb(.(fly, So), So, s(p, N, fly)).<BR>=(X, X).<BR><BR>Queries:<BR><BR>goal().<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>np_in</font>: (f,f,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>goal_in_</font> &#8594; <FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>) &#8594; <FONT COLOR=#0000cc>U7_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>a</font>)
<BR><FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>the</font>) &#8594; <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>the</font>)
<BR><FONT COLOR=#0000cc>U7_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>s</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>s</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>books</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>p</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>books</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>p</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>s</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>s</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>boxes</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>p</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>boxes</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>p</font>))
<BR><FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U9_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>)) &#8594; <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>P</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>P</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>)) &#8594; <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>P</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>P</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>U9_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>falls</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>falls</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fall</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fall</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>flies</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>flies</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fly</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fly</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>))
<BR><FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_</font></BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_</font>&#160; = &#160;<FONT COLOR=#0000cc>goal_in_</font><BR>
<FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>np_in_aaa</font><BR>
<FONT COLOR=#0000cc>U7_aaa</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_aaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>det_in_aaa</font><BR>
<FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noun_in_aaa</font><BR>
<FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_aaa</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>U9_aaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</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>s</font>&#160; = &#160;<FONT COLOR=#0000cc>s</font><BR>
<FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>the</font>&#160; = &#160;<FONT COLOR=#0000cc>the</font><BR>
<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>verb_out_aag</font><BR>
<FONT COLOR=#0000cc>p</font>&#160; = &#160;<FONT COLOR=#0000cc>p</font><BR>
<FONT COLOR=#0000cc>goal_out_</font>&#160; = &#160;<FONT COLOR=#0000cc>goal_out_</font><BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 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>goal_in_</font> &#8594; <FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>) &#8594; <FONT COLOR=#0000cc>U7_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>a</font>)
<BR><FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>the</font>) &#8594; <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>the</font>)
<BR><FONT COLOR=#0000cc>U7_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>s</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>s</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>books</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>p</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>books</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>p</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>s</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>s</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>boxes</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>p</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>boxes</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>p</font>))
<BR><FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U9_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>)) &#8594; <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>P</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>P</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>)) &#8594; <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>P</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>P</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>U9_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>falls</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>falls</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fall</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fall</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>flies</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>flies</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fly</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fly</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>))
<BR><FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_</font></BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_</font>&#160; = &#160;<FONT COLOR=#0000cc>goal_in_</font><BR>
<FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>np_in_aaa</font><BR>
<FONT COLOR=#0000cc>U7_aaa</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_aaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>det_in_aaa</font><BR>
<FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noun_in_aaa</font><BR>
<FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_aaa</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>U9_aaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</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>s</font>&#160; = &#160;<FONT COLOR=#0000cc>s</font><BR>
<FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>the</font>&#160; = &#160;<FONT COLOR=#0000cc>the</font><BR>
<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>verb_out_aag</font><BR>
<FONT COLOR=#0000cc>p</font>&#160; = &#160;<FONT COLOR=#0000cc>p</font><BR>
<FONT COLOR=#0000cc>goal_out_</font>&#160; = &#160;<FONT COLOR=#0000cc>goal_out_</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>GOAL_IN_</font> &#8594; <FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>GOAL_IN_</font> &#8594; <FONT COLOR=#0000cc>NP_IN_AAA</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>NP_IN_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>) &#8594; <FONT COLOR=#0000cc>U7_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>NP_IN_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>) &#8594; <FONT COLOR=#0000cc>DET_IN_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U7_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U7_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>NOUN_IN_AAA</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U9_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>COMB_IN_GGA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U2_<SUP>1</SUP></font>(<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>VERB_IN_AAG</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>goal_in_</font> &#8594; <FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>) &#8594; <FONT COLOR=#0000cc>U7_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>a</font>)
<BR><FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>the</font>) &#8594; <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>the</font>)
<BR><FONT COLOR=#0000cc>U7_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>s</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>s</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>books</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>p</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>books</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>p</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>s</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>s</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>boxes</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>p</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>boxes</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>p</font>))
<BR><FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U9_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>)) &#8594; <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>P</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>P</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>)) &#8594; <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>P</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>P</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>U9_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>falls</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>falls</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fall</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fall</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>flies</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>flies</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fly</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fly</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>))
<BR><FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_</font></BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_</font>&#160; = &#160;<FONT COLOR=#0000cc>goal_in_</font><BR>
<FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>np_in_aaa</font><BR>
<FONT COLOR=#0000cc>U7_aaa</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_aaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>det_in_aaa</font><BR>
<FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noun_in_aaa</font><BR>
<FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_aaa</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>U9_aaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</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>s</font>&#160; = &#160;<FONT COLOR=#0000cc>s</font><BR>
<FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>the</font>&#160; = &#160;<FONT COLOR=#0000cc>the</font><BR>
<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>verb_out_aag</font><BR>
<FONT COLOR=#0000cc>p</font>&#160; = &#160;<FONT COLOR=#0000cc>p</font><BR>
<FONT COLOR=#0000cc>goal_out_</font>&#160; = &#160;<FONT COLOR=#0000cc>goal_out_</font><BR>
<FONT COLOR=#0000cc>DET_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DET_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U9_AAA</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>U9_AAA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_AAA</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_AAA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_<SUP>1</SUP></font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_<SUP>1</SUP></font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>VERB_IN_AAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>VERB_IN_AAG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>NP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>NP_IN_AAA</font><BR>
<FONT COLOR=#0000cc>GOAL_IN_</font>&#160; = &#160;<FONT COLOR=#0000cc>GOAL_IN_</font><BR>
<FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>COMB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>COMB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>NOUN_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>NOUN_IN_AAA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PredefinedPredicateTransformerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 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>GOAL_IN_</font> &#8594; <FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>GOAL_IN_</font> &#8594; <FONT COLOR=#0000cc>NP_IN_AAA</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>NP_IN_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>) &#8594; <FONT COLOR=#0000cc>U7_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>NP_IN_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>) &#8594; <FONT COLOR=#0000cc>DET_IN_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U7_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U7_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>NOUN_IN_AAA</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U9_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>COMB_IN_GGA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U2_<SUP>1</SUP></font>(<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>VERB_IN_AAG</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>goal_in_</font> &#8594; <FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>) &#8594; <FONT COLOR=#0000cc>U7_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>a</font>)
<BR><FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>the</font>) &#8594; <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>the</font>)
<BR><FONT COLOR=#0000cc>U7_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>s</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>s</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>books</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>p</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>books</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>book</font>, <FONT COLOR=#0000cc>p</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>s</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>s</font>))
<BR><FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>boxes</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>p</font>)) &#8594; <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>boxes</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>box</font>, <FONT COLOR=#0000cc>p</font>))
<BR><FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>St</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U9_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>)) &#8594; <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>P</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>P</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>)) &#8594; <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#0000cc>the</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>P</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>P</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>U9_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>Si</font>, <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>falls</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>falls</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fall</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fall</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fall</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>flies</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>flies</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>))
<BR><FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fly</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>)) &#8594; <FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>fly</font>, <FONT COLOR=#cc0000>So</font>), <FONT COLOR=#cc0000>So</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>p</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>fly</font>))
<BR><FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>S3</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_</font></BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_</font>&#160; = &#160;<FONT COLOR=#0000cc>goal_in_</font><BR>
<FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>np_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>np_in_aaa</font><BR>
<FONT COLOR=#0000cc>U7_aaa</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_aaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>det_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>det_in_aaa</font><BR>
<FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>det_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_aaa</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>noun_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noun_in_aaa</font><BR>
<FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>noun_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_aaa</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>U9_aaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>comb_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</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>s</font>&#160; = &#160;<FONT COLOR=#0000cc>s</font><BR>
<FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>comb_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>the</font>&#160; = &#160;<FONT COLOR=#0000cc>the</font><BR>
<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>np_out_aaa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>verb_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>verb_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>verb_out_aag</font><BR>
<FONT COLOR=#0000cc>p</font>&#160; = &#160;<FONT COLOR=#0000cc>p</font><BR>
<FONT COLOR=#0000cc>goal_out_</font>&#160; = &#160;<FONT COLOR=#0000cc>goal_out_</font><BR>
<FONT COLOR=#0000cc>DET_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>DET_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U9_AAA</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>U9_AAA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_AAA</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_AAA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_<SUP>1</SUP></font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_<SUP>1</SUP></font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>VERB_IN_AAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>VERB_IN_AAG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_AAA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>NP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>NP_IN_AAA</font><BR>
<FONT COLOR=#0000cc>GOAL_IN_</font>&#160; = &#160;<FONT COLOR=#0000cc>GOAL_IN_</font><BR>
<FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_<SUP>1</SUP></font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>COMB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>COMB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>NOUN_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>NOUN_IN_AAA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 0 SCCs with 10 less nodes.<BR><BR></body>

