YES
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Frameset//EN"
"http:/www.w3.org/TR/html4/frameset.dtd">
<html>
<head>
<title>Left Termination proof of ../tpdb/LP/BCGGV05/der-bf.pl</title>
</head>
<body>
<BR><B>Left Termination</B> of the query pattern
p_in_2(g, a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>p(d(e(t)), const(1)).<BR>p(d(e(const(A))), const(0)).<BR>p(d(e(+(X, Y))), +(DX, DY))&#160;:-&#160;','(p(d(e(X)), DX), p(d(e(Y)), DY)).<BR>p(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))&#160;:-&#160;','(p(d(e(X)), DX), p(d(e(Y)), DY)).<BR>p(d(d(X)), DDX)&#160;:-&#160;','(p(d(X), DX), p(d(e(DX)), DDX)).<BR><BR>Queries:<BR><BR>p(g,a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>p_in</font>: (b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>t</font>&#160; = &#160;<FONT COLOR=#0000cc>t</font><BR>
<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>const</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>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>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>t</font>&#160; = &#160;<FONT COLOR=#0000cc>t</font><BR>
<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>const</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>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>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>t</font>&#160; = &#160;<FONT COLOR=#0000cc>t</font><BR>
<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>const</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>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>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>x1</font>, <FONT COLOR=#cc0000>x2</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>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>t</font>&#160; = &#160;<FONT COLOR=#0000cc>t</font><BR>
<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>const</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>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>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>x1</font>, <FONT COLOR=#cc0000>x2</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 2 SCCs with 5 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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>t</font>&#160; = &#160;<FONT COLOR=#0000cc>t</font><BR>
<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>const</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>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>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>t</font>&#160; = &#160;<FONT COLOR=#0000cc>t</font><BR>
<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>const</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>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>x1</font>, <FONT COLOR=#cc0000>x2</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>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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 QDPOrderProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>))) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>)))) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>))))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>))))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_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>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>*</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>+</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>1</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>P_IN_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>const</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>d</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>e</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_in_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_out_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>t</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
none<BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 DependencyGraphProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>))) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>)))) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>))))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>))))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_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>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 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 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 DependencyGraphProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 QDPOrderProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>))) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>)))) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>))))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>))))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_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>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>*</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>+</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>1</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>P_IN_GA</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_GA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>const</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>d</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>e</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_in_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_out_ga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>t</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
none<BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 DependencyGraphProof</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 QDPOrderProof</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>))) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>)))) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>))))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>))))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_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>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.<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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>t</font>)), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>1</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>A</font>))), <FONT COLOR=#0000cc>const</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>))) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DDX</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>DX</font>)), <FONT COLOR=#cc0000>DDX</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DY</font>), <FONT COLOR=#0000cc>*</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>)))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DX</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>, <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>Y</font>)), <FONT COLOR=#cc0000>DY</font>)) &#8594; <FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))), <FONT COLOR=#0000cc>+</font>(<FONT COLOR=#cc0000>DX</font>, <FONT COLOR=#cc0000>DY</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>e</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>t</font>&#160; = &#160;<FONT COLOR=#0000cc>t</font><BR>
<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>const</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>const</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</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>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>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>DDX</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>DX</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#cc0000>x1</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><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</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>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GA</font>(<FONT COLOR=#0000cc>d</font>(<FONT COLOR=#cc0000>X</font>))<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR></body>

