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<BR><B>Left Termination</B> of the query pattern
append_in_3(a, a, a)
w.r.t. the given <I>Prolog program</I> could not be shown:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>append([], L, L).<BR>append(.(H, L1), L2, .(H, L3))&#160;:-&#160;append(L1, L2, L3).<BR>append1([], L, L).<BR>append1(.(H, L1), L2, .(H, L3))&#160;:-&#160;append1(L1, L2, L3).<BR><BR>Queries:<BR><BR>append(a,a,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>append_in</font>: (f,f,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>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</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>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_AAA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AAA</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_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_AAA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_AAA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AAA</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_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_AAA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] 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 <B>PiDP</B></pre><pre>              &#8627 UsableRulesProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_AAA</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 PiDP</pre><pre>              &#8627 UsableRulesProof</pre><pre>                &#8627 <B>PiDP</B></pre><pre>                  &#8627 PiDPToQDPProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_AAA</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 UsableRulesProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 PiDPToQDPProof</pre><pre>                    &#8627 <B>QDP</B></pre><pre>                      &#8627 NonTerminationProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font></BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by semiunifying a rule from P directly.<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>APPEND_IN_AAA</font> evaluates to  t =<FONT COLOR=#0000cc>APPEND_IN_AAA</font><BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR>The DP semiunifies directly so there is only one rewrite step from APPEND_IN_AAA to APPEND_IN_AAA.<BR><BR><BR><BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>append_in</font>: (f,f,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>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</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>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_AAA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AAA</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_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_AAA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_AAA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AAA</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_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_AAA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 1 SCC with 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 <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>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U1_aaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>append_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_aaa</font><BR>
<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_aaa</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_aaa</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>x2</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_AAA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 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>APPEND_IN_AAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_AAA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 UsableRulesProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 PiDPToQDPProof</pre><pre>                    &#8627 <B>QDP</B></pre><pre>                      &#8627 NonTerminationProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font></BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by semiunifying a rule from P directly.<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APPEND_IN_AAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>APPEND_IN_AAA</font> evaluates to  t =<FONT COLOR=#0000cc>APPEND_IN_AAA</font><BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR>The DP semiunifies directly so there is only one rewrite step from APPEND_IN_AAA to APPEND_IN_AAA.<BR><BR><BR><BR><BR><BR></body>

