YES
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<BR><B>Left Termination</B> of the query pattern
at_in_2(a, 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>at(X, fido)&#160;:-&#160;','(at(X, mary), near(X)).<BR>at(ta, mary).<BR>at(jm, mary).<BR>near(jm).<BR><BR>Queries:<BR><BR>at(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>at_in</font>: (f,f) (f,b)
<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>at_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#0000cc>jm</font>) &#8594; <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#0000cc>jm</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>)
<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>fido</font>&#160; = &#160;<FONT COLOR=#0000cc>fido</font><BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mary</font>&#160; = &#160;<FONT COLOR=#0000cc>mary</font><BR>
<FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>jm</font>&#160; = &#160;<FONT COLOR=#0000cc>jm</font><BR>
<FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>near_out_g</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>x1</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><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#0000cc>jm</font>) &#8594; <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#0000cc>jm</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>)
<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>fido</font>&#160; = &#160;<FONT COLOR=#0000cc>fido</font><BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mary</font>&#160; = &#160;<FONT COLOR=#0000cc>mary</font><BR>
<FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>jm</font>&#160; = &#160;<FONT COLOR=#0000cc>jm</font><BR>
<FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>near_out_g</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>x1</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>AT_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>AT_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>NEAR_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>NEAR_IN_G</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#0000cc>jm</font>) &#8594; <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#0000cc>jm</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>)
<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>fido</font>&#160; = &#160;<FONT COLOR=#0000cc>fido</font><BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mary</font>&#160; = &#160;<FONT COLOR=#0000cc>mary</font><BR>
<FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>jm</font>&#160; = &#160;<FONT COLOR=#0000cc>jm</font><BR>
<FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>near_out_g</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>NEAR_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>NEAR_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>AT_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>AT_IN_AA</font><BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</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>AT_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>AT_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>NEAR_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>NEAR_IN_G</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>))
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#0000cc>jm</font>) &#8594; <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#0000cc>jm</font>)
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mary</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>fido</font>)
<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#0000cc>ta</font>, <FONT COLOR=#0000cc>mary</font>)
<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>) &#8594; <FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#0000cc>jm</font>, <FONT COLOR=#0000cc>mary</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>at_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>fido</font>&#160; = &#160;<FONT COLOR=#0000cc>fido</font><BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mary</font>&#160; = &#160;<FONT COLOR=#0000cc>mary</font><BR>
<FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>near_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>jm</font>&#160; = &#160;<FONT COLOR=#0000cc>jm</font><BR>
<FONT COLOR=#0000cc>near_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>near_out_g</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>at_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>AT_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>NEAR_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>NEAR_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>AT_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>AT_IN_AA</font><BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 0 SCCs with 8 less nodes.<BR><BR></body>

