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<title>Left Termination proof of ../tpdb/LP/talp/plumer/pl4.5.2.pl</title>
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<BR><B>Left Termination</B> of the query pattern
s2_in_2(g, a)
w.r.t. the given <I>Prolog program</I> could not be shown:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>s2(plus(A, plus(B, C)), D)&#160;:-&#160;s2(plus(plus(A, B), C), D).<BR>s2(plus(A, B), C)&#160;:-&#160;s2(plus(B, A), C).<BR>s2(plus(X, 0), X).<BR>s2(plus(X, Y), Z)&#160;:-&#160;','(s2(X, A), ','(s2(Y, B), s2(plus(A, B), Z))).<BR>s2(plus(A, B), C)&#160;:-&#160;','(isNat(A), ','(isNat(B), add(A, B, C))).<BR>isNat(s(X))&#160;:-&#160;isNat(X).<BR>isNat(0).<BR>add(s(X), Y, s(Z))&#160;:-&#160;add(X, Y, Z).<BR>add(0, X, X).<BR><BR>Queries:<BR><BR>s2(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>s2_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>isNat_in</font>: (b)
<BR><FONT COLOR=#0000cc>add_in</font>: (b,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_g</font><BR>
<FONT COLOR=#0000cc>U7_ga</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_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_g</font><BR>
<FONT COLOR=#0000cc>U7_ga</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_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x4</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>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_g</font><BR>
<FONT COLOR=#0000cc>U7_ga</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_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_GGA</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>U10_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISNAT_IN_G</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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</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>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</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>U5_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U9_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_G</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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_g</font><BR>
<FONT COLOR=#0000cc>U7_ga</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_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_GGA</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>U10_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISNAT_IN_G</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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</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>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</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>U5_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U9_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_G</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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 3 SCCs with 11 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_g</font><BR>
<FONT COLOR=#0000cc>U7_ga</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_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 >= 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_g</font><BR>
<FONT COLOR=#0000cc>U7_ga</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_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>R is empty.<BR>Pi is empty.<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 PiDPToQDPProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_g</font><BR>
<FONT COLOR=#0000cc>U7_ga</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_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</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>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 <B>QDP</B></pre><pre>                    &#8627 QDPOrderProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_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>U2_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>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>S2_IN_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U10_gga</font>(x<SUB>1</SUB>)</B>) = 0<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>U3_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_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>U4_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>U4_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_ga</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_ga</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>add_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>add_out_gga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isNat_in_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isNat_out_g</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>plus</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s2_in_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s2_out_ga</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>D</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 DependencyGraphProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_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>U2_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 1 SCC with 2 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 DependencyGraphProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 UsableRulesProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_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>U2_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>As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 DependencyGraphProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 UsableRulesProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 QReductionProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))</BLOCKQUOTE><BR>R is empty.<BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_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>U2_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>We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_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>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 DependencyGraphProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 UsableRulesProof</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 QReductionProof</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 NonTerminationProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by semiunifying a rule from P directly.<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) evaluates to  t =<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))<BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Matcher: [<FONT COLOR=#cc0000>B</font> / <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>A</font> / <FONT COLOR=#cc0000>B</font>]</LI>
<LI> Semiunifier: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR>The DP semiunifies directly so there is only one rewrite step from S2_IN_GA(plus(A, B)) to S2_IN_GA(plus(B, A)).<BR><BR><BR><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>s2_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>isNat_in</font>: (b)
<BR><FONT COLOR=#0000cc>add_in</font>: (b,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</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>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_GGA</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>U10_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISNAT_IN_G</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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</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>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</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>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U9_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_G</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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_GGA</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>U10_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISNAT_IN_G</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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</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>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</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>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U9_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_G</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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 3 SCCs with 11 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 >= 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>R is empty.<BR>Pi is empty.<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>ISNAT_IN_G</font>(<FONT COLOR=#cc0000>X</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <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>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>isNat_out_g</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=#cc0000>C</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>isNat_out_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>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U10_gga</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>U10_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</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>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 <B>QDP</B></pre><pre>                    &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</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>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>S2_IN_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U10_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>3</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>3</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_g</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>add_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>add_out_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isNat_in_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isNat_out_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>plus</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s2_in_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s2_out_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</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>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</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>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>S2_IN_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U10_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = x<SUB>3</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>3</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + 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>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_g</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>add_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>add_out_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isNat_in_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>isNat_out_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>plus</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s2_in_ga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s2_out_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</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>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</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>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 DependencyGraphProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 UsableRulesProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>)))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>isNat_out_g</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>add_in_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>s2_out_ga</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>D</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 DependencyGraphProof</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 UsableRulesProof</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 QReductionProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))</BLOCKQUOTE><BR>R is empty.<BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>s2_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>isNat_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U10_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 DependencyGraphProof</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 UsableRulesProof</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 QReductionProof</pre><pre>                                      &#8627 <B>QDP</B></pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))) &#8594; <FONT COLOR=#0000cc>S2_IN_GA</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#0000cc>plus</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>))</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR><BR></body>

