YES
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<title>Left Termination proof of ../tpdb/LP/lpexamples/shapes.pl</title>
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<BR><B>Left Termination</B> of the query pattern
shapes_in_2(g, a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>shapes(Matrix, N)&#160;:-&#160;','(varmat(Matrix, MatrixWithVars), unif_matrx(MatrixWithVars)).<BR>varmat([], []).<BR>varmat(.(L, Ls), .(VL, VLs))&#160;:-&#160;','(varmat(L, VL), varmat(Ls, VLs)).<BR>varmat(.(black, Xs), .(black, VXs))&#160;:-&#160;varmat(Xs, VXs).<BR>varmat(.(white, Xs), .(w(X), VXs))&#160;:-&#160;varmat(Xs, VXs).<BR>unif_matrx(.(L1, .(L2, Ls)))&#160;:-&#160;','(unif_lines(L1, L2), unif_matrx(.(L2, Ls))).<BR>unif_matrx(.(X, [])).<BR>unif_lines(.(W, .(X, L1s)), .(Y, .(Z, L2s)))&#160;:-&#160;','(unif_pairs(.(W, .(X, .(Y, .(Z, .(W, .(Y, .(X, .(Z, .(W, .(Z, .(X, .(Y, []))))))))))))), unif_lines(.(X, L1s), .(Z, L2s))).<BR>unif_lines(.(X, []), .(X1, [])).<BR>unif_pairs([]).<BR>unif_pairs(.(A, .(B, Pairs)))&#160;:-&#160;','(unif(A, B), unif_pairs(Pairs)).<BR>unif(w(A), w(A)).<BR>unif(black, black).<BR>unif(black, w(X)).<BR>unif(w(X), black).<BR><BR>Queries:<BR><BR>shapes(g,a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>shapes_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>varmat_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>unif_matrx_in</font>: (b)
<BR><FONT COLOR=#0000cc>unif_lines_in</font>: (b,b)
<BR><FONT COLOR=#0000cc>unif_pairs_in</font>: (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>shapes_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>white</font>&#160; = &#160;<FONT COLOR=#0000cc>white</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</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>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</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>U9_gg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</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>U10_gg</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_out_gg</font><BR>
<FONT COLOR=#0000cc>U8_g</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>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_out_g</font><BR>
<FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_out_ga</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>shapes_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>white</font>&#160; = &#160;<FONT COLOR=#0000cc>white</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</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>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</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>U9_gg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</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>U10_gg</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_out_gg</font><BR>
<FONT COLOR=#0000cc>U8_g</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>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_out_g</font><BR>
<FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_out_ga</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>SHAPES_IN_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>SHAPES_IN_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)
<BR><FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)
<BR><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))
<BR><FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>UNIF_IN_GG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>Pairs</font>)
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>white</font>&#160; = &#160;<FONT COLOR=#0000cc>white</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</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>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</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>U9_gg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</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>U10_gg</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_out_gg</font><BR>
<FONT COLOR=#0000cc>U8_g</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>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_out_g</font><BR>
<FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_G</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>U7_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SHAPES_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SHAPES_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U12_G</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>U12_G</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_GG</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>U10_GG</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>UNIF_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</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>U9_GG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_G</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>U8_G</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_G</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>U11_G</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>x1</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>SHAPES_IN_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>SHAPES_IN_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)
<BR><FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)
<BR><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))
<BR><FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>UNIF_IN_GG</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>Pairs</font>)
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>white</font>&#160; = &#160;<FONT COLOR=#0000cc>white</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</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>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</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>U9_gg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</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>U10_gg</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_out_gg</font><BR>
<FONT COLOR=#0000cc>U8_g</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>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_out_g</font><BR>
<FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_G</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>U7_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SHAPES_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SHAPES_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U12_G</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>U12_G</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U10_GG</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>U10_GG</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>UNIF_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</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>U9_GG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U8_G</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>U8_G</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U11_G</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>U11_G</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 4 SCCs with 13 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><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>Pairs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>white</font>&#160; = &#160;<FONT COLOR=#0000cc>white</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</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>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</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>U9_gg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</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>U10_gg</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_out_gg</font><BR>
<FONT COLOR=#0000cc>U8_g</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>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_out_g</font><BR>
<FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_out_ga</font><BR>
<FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U11_G</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>U11_G</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>Pairs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U11_G</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>U11_G</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>) &#8594; <FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>Pairs</font>)
<BR><FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>) &#8594; <FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#cc0000>Pairs</font>)<BR>The graph contains the following edges 1 >= 1<P></LI>
<LI><FONT COLOR=#0000cc>UNIF_PAIRS_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_G</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</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><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))
<BR><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>white</font>&#160; = &#160;<FONT COLOR=#0000cc>white</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</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>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</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>U9_gg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</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>U10_gg</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_out_gg</font><BR>
<FONT COLOR=#0000cc>U8_g</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>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_out_g</font><BR>
<FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_out_ga</font><BR>
<FONT COLOR=#0000cc>U9_GG</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>U9_GG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>x1</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 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))
<BR><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</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>U9_GG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 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 Rewriting</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) at position [4] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))), <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</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 Rewriting</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 QDPOrderProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))
<BR><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))), <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x0</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>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))), <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U11_g</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U12_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 1 + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>[]</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>black</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_pairs_in_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_pairs_out_g</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>w</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
none<BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Rewriting</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 QDPOrderProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 DependencyGraphProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>UNIF_LINES_IN_GG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</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>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))
<BR><FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>white</font>&#160; = &#160;<FONT COLOR=#0000cc>white</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</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>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</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>U9_gg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</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>U10_gg</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_out_gg</font><BR>
<FONT COLOR=#0000cc>U8_g</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>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_out_g</font><BR>
<FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_out_ga</font><BR>
<FONT COLOR=#0000cc>U7_G</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>U7_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 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>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))
<BR><FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</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>U9_gg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</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>U10_gg</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_out_gg</font><BR>
<FONT COLOR=#0000cc>U7_G</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>U7_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPOrderProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>) &#8594; <FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))
<BR><FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>unif_lines_out_gg</font>) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x0</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>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>) &#8594; <FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U10_gg</font>(x<SUB>1</SUB>)</B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U11_g</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U12_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_G</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>[]</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>black</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_lines_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_lines_out_gg</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_out_gg</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_pairs_in_g</font>(x<SUB>1</SUB>)</B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>unif_pairs_out_g</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>w</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>unif_lines_out_gg</font>) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</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 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 DependencyGraphProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>) &#8594; <FONT COLOR=#0000cc>UNIF_MATRX_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>unif_lines_out_gg</font>) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#0000cc>unif_pairs_out_g</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</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>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>))))))))))))))
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>))) &#8594; <FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>A</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>) &#8594; <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>black</font>)
<BR><FONT COLOR=#0000cc>U11_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>)) &#8594; <FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>Pairs</font>))
<BR><FONT COLOR=#0000cc>U12_g</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>Pairs</font>)) &#8594; <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>Pairs</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>[]</font>)))))))))))))) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>))) &#8594; <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L1s</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>L2s</font>)))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>))) &#8594; <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>Ls</font>)))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>MatrixWithVars</font>)) &#8594; <FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>Matrix</font>, <FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>white</font>&#160; = &#160;<FONT COLOR=#0000cc>white</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</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>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</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>U9_gg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>unif_pairs_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_pairs_out_g</font><BR>
<FONT COLOR=#0000cc>U11_g</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>U11_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>unif_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_out_gg</font><BR>
<FONT COLOR=#0000cc>U12_g</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>U12_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</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>U10_gg</font>(<FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>unif_lines_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_lines_out_gg</font><BR>
<FONT COLOR=#0000cc>U8_g</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>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>unif_matrx_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>unif_matrx_out_g</font><BR>
<FONT COLOR=#0000cc>shapes_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>shapes_out_ga</font><BR>
<FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</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>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>VXs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>black</font>&#160; = &#160;<FONT COLOR=#0000cc>black</font><BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>white</font>&#160; = &#160;<FONT COLOR=#0000cc>white</font><BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>w</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>w</font><BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Ls</font>)
<BR><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>Ls</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>VXs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>w</font>, <FONT COLOR=#cc0000>VXs</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>VLs</font>)) &#8594; <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VL</font>, <FONT COLOR=#cc0000>VLs</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>varmat_in_ga</font>(<FONT COLOR=#cc0000>L</font>))<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Ls</font>, <FONT COLOR=#0000cc>varmat_out_ga</font>(<FONT COLOR=#cc0000>VL</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Ls</font>)<BR>The graph contains the following edges 1 >= 1<P></LI>
<LI><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>Ls</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>L</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>black</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>white</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>VARMAT_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR></body>

