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<title>Left Termination proof of ../tpdb/LP/lpexamples/ackermann.pl</title>
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<BR><B>Left Termination</B> of the query pattern
ackermann_in_3(g, 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>ackermann(0, N, s(N)).<BR>ackermann(s(M), 0, Val)&#160;:-&#160;ackermann(M, s(0), Val).<BR>ackermann(s(M), s(N), Val)&#160;:-&#160;','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val)).<BR><BR>Queries:<BR><BR>ackermann(g,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>ackermann_in</font>: (b,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>))
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>))
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 1 SCC with 2 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 <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>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>, <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Val</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 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>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>)
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)))
<BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>Val</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>Val</font>)) &#8594; <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>Val</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>ackermann_in_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>))<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>ackermann_out_gga</font>(<FONT COLOR=#cc0000>Val1</font>)) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>Val1</font>)<BR>The graph contains the following edges 1 >= 1, 2 > 2<P></LI>
<LI><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#cc0000>N</font>)<BR>The graph contains the following edges 1 >= 1, 2 > 2<P></LI>
<LI><FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>ACKERMANN_IN_GGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR></body>

