YES
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<title>Left Termination proof of ../tpdb/LP/talp/plumer/pl8.4.2.pl</title>
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<BR><B>Left Termination</B> of the query pattern
e_in_2(g, a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>e(L, T)&#160;:-&#160;t(L, T).<BR>e(L, T)&#160;:-&#160;','(t(L, .(plus, C)), e(C, T)).<BR>t(L, T)&#160;:-&#160;n(L, T).<BR>t(L, T)&#160;:-&#160;','(n(L, .(star, C)), t(C, T)).<BR>n(.(L, T), T)&#160;:-&#160;z(L).<BR>n(.(lbrace, A), B)&#160;:-&#160;e(A, .(rbrace, B)).<BR>z(a).<BR>z(b).<BR>z(c).<BR><BR>Queries:<BR><BR>e(g,a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>e_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>t_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>n_in</font>: (b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>a</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>c</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>c</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_out_g</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>c</font>&#160; = &#160;<FONT COLOR=#0000cc>c</font><BR>
<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>lbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>lbrace</font><BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>star</font>&#160; = &#160;<FONT COLOR=#0000cc>star</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>plus</font>&#160; = &#160;<FONT COLOR=#0000cc>plus</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>rbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>rbrace</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>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>a</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>c</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>c</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_out_g</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>c</font>&#160; = &#160;<FONT COLOR=#0000cc>c</font><BR>
<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>lbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>lbrace</font><BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>star</font>&#160; = &#160;<FONT COLOR=#0000cc>star</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>plus</font>&#160; = &#160;<FONT COLOR=#0000cc>plus</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>rbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>rbrace</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>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>Z_IN_G</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>a</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>c</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>c</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_out_g</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>c</font>&#160; = &#160;<FONT COLOR=#0000cc>c</font><BR>
<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>lbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>lbrace</font><BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>star</font>&#160; = &#160;<FONT COLOR=#0000cc>star</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>plus</font>&#160; = &#160;<FONT COLOR=#0000cc>plus</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>rbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>rbrace</font><BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>Z_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>Z_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>Z_IN_G</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>a</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>c</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>c</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_out_g</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>c</font>&#160; = &#160;<FONT COLOR=#0000cc>c</font><BR>
<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>lbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>lbrace</font><BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>star</font>&#160; = &#160;<FONT COLOR=#0000cc>star</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>plus</font>&#160; = &#160;<FONT COLOR=#0000cc>plus</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>rbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>rbrace</font><BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>Z_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>Z_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 1 SCC with 7 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 <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>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>a</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>c</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#0000cc>c</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>z_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>z_out_g</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>c</font>&#160; = &#160;<FONT COLOR=#0000cc>c</font><BR>
<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>lbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>lbrace</font><BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>star</font>&#160; = &#160;<FONT COLOR=#0000cc>star</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>plus</font>&#160; = &#160;<FONT COLOR=#0000cc>plus</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>rbrace</font>&#160; = &#160;<FONT COLOR=#0000cc>rbrace</font><BR>
<FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<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 PiDP</pre><pre>              &#8627 PiDPToQDPProof</pre><pre>                &#8627 <B>QDP</B></pre><pre>                  &#8627 UsableRulesReductionPairsProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>c</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_out_g</font>) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.<BR><BR>The following dependency pairs can be deleted:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>E_IN_GA</font>(<FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE>The following rules are removed from R:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>lbrace</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>star</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>rbrace</font>, <FONT COLOR=#cc0000>B</font>))) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>plus</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>a</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font>
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#0000cc>c</font>) &#8594; <FONT COLOR=#0000cc>z_out_g</font></BLOCKQUOTE>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2 + x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>E_IN_GA</font>(x<SUB>1</SUB>)</B>) = 2 + 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>N_IN_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>T_IN_GA</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_ga</font>(x<SUB>1</SUB>)</B>) = 1 + 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_ga</font>(x<SUB>1</SUB>)</B>) = 2 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_ga</font>(x<SUB>1</SUB>)</B>) = 2 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>a</font></B>) = 2<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>b</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>c</font></B>) = 2<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>e_in_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>e_out_ga</font>(x<SUB>1</SUB>)</B>) = 2 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>lbrace</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>n_in_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>n_out_ga</font>(x<SUB>1</SUB>)</B>) = 2 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>plus</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>rbrace</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>star</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>t_in_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>t_out_ga</font>(x<SUB>1</SUB>)</B>) = 2 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>z_in_g</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>z_out_g</font></B>) = 2<sup>&nbsp;</sup> <sub>&nbsp;</sub></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 PiDP</pre><pre>              &#8627 PiDPToQDPProof</pre><pre>                &#8627 QDP</pre><pre>                  &#8627 UsableRulesReductionPairsProof</pre><pre>                    &#8627 <B>QDP</B></pre><pre>                      &#8627 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>N_IN_GA</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>T_IN_GA</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>e_out_ga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>z_out_g</font>) &#8594; <FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>n_out_ga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>t_out_ga</font>(<FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>e_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>t_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>n_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>z_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.<BR><BR></body>

