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<title>Left Termination proof of ../tpdb/LP/talp/talp/evaluate.pl</title>
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<BR><B>Left Termination</B> of the query pattern
myis_in_2(a, g)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>myis(Z, X)&#160;:-&#160;evaluate(X, Z).<BR>evaluate(+(X, Y), Z)&#160;:-&#160;','(evaluate(X, X1), ','(evaluate(Y, Y1), add(X1, Y1, Z))).<BR>evaluate(-(X, Y), Z)&#160;:-&#160;','(evaluate(X, X1), ','(evaluate(Y, Y1), sub(X1, Y1, Z))).<BR>evaluate(*(X, Y), Z)&#160;:-&#160;','(evaluate(X, X1), ','(evaluate(Y, Y1), mult(X1, Y1, Z))).<BR>evaluate(X, X)&#160;:-&#160;integer(X).<BR>integer(s(X))&#160;:-&#160;integer(X).<BR>integer(0).<BR>add(s(X), Y, s(Z))&#160;:-&#160;add(X, Y, Z).<BR>add(0, X, X).<BR>sub(s(X), s(Y), Z)&#160;:-&#160;sub(X, Y, Z).<BR>sub(X, 0, X).<BR>mult(s(X), Y, R)&#160;:-&#160;','(mult(X, Y, Z), add(Y, Z, R)).<BR>mult(0, Y, 0).<BR>notEq(s(X), s(Y))&#160;:-&#160;notEq(X, Y).<BR>notEq(s(X), 0).<BR>notEq(0, s(X)).<BR>lt(s(X), s(Y))&#160;:-&#160;lt(X, Y).<BR>lt(0, s(Y)).<BR>gt(s(X), s(Y))&#160;:-&#160;gt(X, Y).<BR>gt(s(X), 0).<BR>le(s(X), s(Y))&#160;:-&#160;le(X, Y).<BR>le(0, s(Y)).<BR>le(0, 0).<BR><BR>Queries:<BR><BR>myis(a,g).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>myis_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>evaluate_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>integer_in</font>: (b)
<BR><FONT COLOR=#0000cc>mult_in</font>: (b,b,f)
<BR><FONT COLOR=#0000cc>add_in</font>: (b,b,f)
<BR><FONT COLOR=#0000cc>sub_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>myis_in_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U13_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>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U13_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>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MYIS_IN_AG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>MYIS_IN_AG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U9_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U9_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U15_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U15_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>SUB_IN_GGA</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U13_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>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U11_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>x1</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>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_GGA</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>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_GGA</font>(<FONT COLOR=#cc0000>x4</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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U13_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>U13_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>MYIS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MYIS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_GA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MYIS_IN_AG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>MYIS_IN_AG</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U9_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U9_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U15_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U15_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>SUB_IN_GGA</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>ADD_IN_GGA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U13_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>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U11_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>x1</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>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_GGA</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>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_GGA</font>(<FONT COLOR=#cc0000>x4</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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U13_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>U13_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>MYIS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>MYIS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_GA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 5 SCCs with 19 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>SUB_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>myis_in_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U13_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>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>SUB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SUB_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 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>SUB_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>SUB_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>SUB_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SUB_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 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>SUB_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>SUB_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>SUB_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>SUB_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 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>myis_in_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U13_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>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_out_ag</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>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 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 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 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 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>MULT_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>myis_in_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U13_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>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MULT_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 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>MULT_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>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MULT_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 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 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>MULT_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>MULT_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>MULT_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 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U13_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>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>INTEGER_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>INTEGER_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 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>INTEGER_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INTEGER_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 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>INTEGER_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INTEGER_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>INTEGER_IN_G</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INTEGER_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 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</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>U13_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>U13_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>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</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>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>myis_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_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>x2</font>, <FONT COLOR=#cc0000>x4</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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</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>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</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=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</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=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</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=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>0</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=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</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>U13_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>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U13_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>U14_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>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>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_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>U12_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>integer_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>integer_out_g</font><BR>
<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_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>U9_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_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>U10_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U15_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>U15_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U16_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>U16_gga</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>U13_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>U13_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>U6_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>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U14_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>U14_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_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>x2</font>, <FONT COLOR=#cc0000>x4</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>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 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><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>integer_out_g</font>) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</font>
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>integer_out_g</font>) &#8594; <FONT COLOR=#0000cc>integer_out_g</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>U13_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>sub_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U13_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>U14_gga</font>(<FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sub_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#0000cc>add_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U11_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</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>)
<BR><FONT COLOR=#0000cc>U9_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>integer_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U10_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>add_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>sub_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U13_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U14_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U15_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U16_gga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><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>U8_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 >= 1<P></LI>
<LI><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 >= 1<P></LI>
<LI><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_out_ga</font>(<FONT COLOR=#cc0000>X1</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 >= 1<P></LI>
<LI><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>))<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U8_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>))<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>evaluate_in_ga</font>(<FONT COLOR=#cc0000>X</font>))<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>EVALUATE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR></body>

