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<title>Left Termination proof of ../tpdb/LP/SGST06/transpose-fb.pl</title>
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<BR><B>Left Termination</B> of the query pattern
transpose_in_2(a, g)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>transpose(A, B)&#160;:-&#160;transpose_aux(A, nil, B).<BR>transpose_aux(cons(R, Rs), X, cons(C, Cs))&#160;:-&#160;','(row2col(R, cons(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1)).<BR>transpose_aux(nil, X, X).<BR>row2col(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))&#160;:-&#160;row2col(Xs, Cols, Cols1, As).<BR>row2col(nil, nil, nil, nil).<BR><BR>Queries:<BR><BR>transpose(a,g).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>transpose_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>transpose_aux_in</font>: (f,b,b)
<BR><FONT COLOR=#0000cc>row2col_in</font>: (f,b,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>transpose_in_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>row2col_in_agaa</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>row2col_in_agaa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_agaa</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>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>row2col_out_agaa</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>row2col_out_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>row2col_in_agaa</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>row2col_in_agaa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_agaa</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>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>row2col_out_agaa</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>row2col_out_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TRANSPOSE_IN_AG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>TRANSPOSE_IN_AG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)
<BR><FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)
<BR><FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>U3_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>))
<BR><FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>row2col_in_agaa</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>row2col_in_agaa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_agaa</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>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>row2col_out_agaa</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>row2col_out_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_AGG</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_AGG</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>TRANSPOSE_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSPOSE_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AGAA</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>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>ROW2COL_IN_AGAA</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>ROW2COL_IN_AGAA</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>TRANSPOSE_IN_AG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>TRANSPOSE_IN_AG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)
<BR><FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)
<BR><FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>U3_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>))
<BR><FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>row2col_in_agaa</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>row2col_in_agaa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_agaa</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>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>row2col_out_agaa</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>row2col_out_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_AGG</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_AGG</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>TRANSPOSE_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSPOSE_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AGAA</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>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>ROW2COL_IN_AGAA</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>ROW2COL_IN_AGAA</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 2 SCCs with 5 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>row2col_in_agaa</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>row2col_in_agaa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_agaa</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>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>row2col_out_agaa</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>row2col_out_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ROW2COL_IN_AGAA</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>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>ROW2COL_IN_AGAA</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>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>)) &#8594; <FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>Cols</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>)) &#8594; <FONT COLOR=#0000cc>ROW2COL_IN_AGAA</font>(<FONT COLOR=#cc0000>Cols</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <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>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>))
<BR><FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)) &#8594; <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_in_agg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_agg</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>row2col_in_agaa</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>row2col_in_agaa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_agaa</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>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>row2col_out_agaa</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>row2col_out_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_agg</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_aux_out_agg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>transpose_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_AGG</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>))
<BR><FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>), <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>row2col_in_agaa</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>row2col_in_agaa</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_agaa</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>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>row2col_out_agaa</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>row2col_out_agaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_AGG</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)))
<BR><FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Cols</font>))
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>Accm</font>)) &#8594; <FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>Accm</font>, <FONT COLOR=#cc0000>Cols1</font>)</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)))</BLOCKQUOTE>Used ordering:  Matrix interpretation [3]:
<BR>Non-tuple symbols: <BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>row2col_in_agaa</font>(x<SUB>1</SUB>)<B> )</B> = </td><td><table><tr><td>/</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>/</td></tr>
</table></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>1</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>nil</font><B> )</B> = </td><td><table><tr><td>/</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>1</td><td>/</td></tr>
</table></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>cons</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)<B> )</B> = </td><td><table><tr><td>/</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>/</td></tr>
</table></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>1</td><td>0</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td><td>+</td><td><table><tr><td>/</td><td>1</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>0</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>2</SUB></B></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>U4_agaa</font>(x<SUB>1</SUB>, ..., x<SUB>3</SUB>)<B> )</B> = </td><td><table><tr><td>/</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>1</td><td>/</td></tr>
</table></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>1</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td><td>+</td><td><table><tr><td>/</td><td>1</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>0</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>2</SUB></B></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>0</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>3</SUB></B></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>row2col_out_agaa</font>(x<SUB>1</SUB>, ..., x<SUB>3</SUB>)<B> )</B> = </td><td><table><tr><td>/</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>/</td></tr>
</table></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>0</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>1</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>0</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>2</SUB></B></td><td>+</td><td><table><tr><td>/</td><td>0</td><td>0</td><td>\</td></tr>
<tr><td>\</td><td>0</td><td>0</td><td>/</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>3</SUB></B></td></tr>
</table><BR>Tuple symbols: <BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)<B> )</B> = </td><td>0</td><td>+</td><td><table><tr><td>[</td><td>0,</td><td>0</td><td>]</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td><td>+</td><td><table><tr><td>[</td><td>0,</td><td>1</td><td>]</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>2</SUB></B></td></tr>
</table><BR><table><tr><td><B>M( </B><FONT COLOR=#0000cc>U2_AGG</font>(x<SUB>1</SUB>)<B> )</B> = </td><td>0</td><td>+</td><td><table><tr><td>[</td><td>1,</td><td>0</td><td>]</td></tr>
</table></td><td>&middot;</td><td><B>x<SUB>1</SUB></B></td></tr>
</table><BR><BR>Matrix type: <BR>We used a basic matrix type which is not further parametrizeable.<BR><BR><BR>As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.<BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Cols</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TRANSPOSE_AUX_IN_AGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGG</font>(<FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>Cs</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>), <FONT COLOR=#cc0000>Cols</font>)) &#8594; <FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>Cols</font>))
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Cols1</font>, <FONT COLOR=#cc0000>As</font>)) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Cols1</font>), <FONT COLOR=#0000cc>cons</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>As</font>))
<BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>row2col_out_agaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>row2col_in_agaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_agaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.<BR><BR></body>

