Left Termination proof of /tmp/tmpIcBsC8/shapes.pl

Left Termination of the query pattern shapes_in_2(g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 NonTerminationProof (⇔)

                ↳15 NO

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 QDPOrderProof (⇔)

                ↳22 QDP

                ↳23 DependencyGraphProof (⇔)

                ↳24 TRUE

            ↳25 PiDP

                ↳26 UsableRulesProof (⇔)

                ↳27 PiDP

                ↳28 PiDPToQDPProof (⇐)

                ↳29 QDP

                ↳30 Narrowing (⇐)

                ↳31 QDP

                ↳32 UsableRulesProof (⇔)

                ↳33 QDP

                ↳34 QReductionProof (⇔)

                ↳35 QDP

                ↳36 NonTerminationProof (⇔)

                ↳37 NO

            ↳38 PiDP

                ↳39 UsableRulesProof (⇔)

                ↳40 PiDP

                ↳41 PiDPToQDPProof (⇐)

                ↳42 QDP

                ↳43 Narrowing (⇐)

                ↳44 QDP

                ↳45 NonTerminationProof (⇔)

                ↳46 NO

            ↳47 PiDP

                ↳48 UsableRulesProof (⇔)

                ↳49 PiDP

                ↳50 PiDPToQDPProof (⇐)

                ↳51 QDP

                ↳52 QDPSizeChangeProof (⇔)

                ↳53 YES

    ↳54 PrologToPiTRSProof (⇐)

        ↳55 PiTRS

        ↳56 DependencyPairsProof (⇔)

        ↳57 PiDP

        ↳58 DependencyGraphProof (⇔)

        ↳59 AND

            ↳60 PiDP

                ↳61 UsableRulesProof (⇔)

                ↳62 PiDP

                ↳63 PiDPToQDPProof (⇐)

                ↳64 QDP

                ↳65 NonTerminationProof (⇔)

                ↳66 NO

            ↳67 PiDP

                ↳68 UsableRulesProof (⇔)

                ↳69 PiDP

                ↳70 PiDPToQDPProof (⇐)

                ↳71 QDP

                ↳72 QDPOrderProof (⇔)

                ↳73 QDP

                ↳74 DependencyGraphProof (⇔)

                ↳75 TRUE

            ↳76 PiDP

                ↳77 UsableRulesProof (⇔)

                ↳78 PiDP

                ↳79 PiDPToQDPProof (⇐)

                ↳80 QDP

                ↳81 Narrowing (⇐)

                ↳82 QDP

                ↳83 UsableRulesProof (⇔)

                ↳84 QDP

                ↳85 QReductionProof (⇔)

                ↳86 QDP

                ↳87 NonTerminationProof (⇔)

                ↳88 NO

            ↳89 PiDP

                ↳90 UsableRulesProof (⇔)

                ↳91 PiDP

                ↳92 PiDPToQDPProof (⇐)

                ↳93 QDP

                ↳94 Narrowing (⇐)

                ↳95 QDP

                ↳96 NonTerminationProof (⇔)

                ↳97 NO

            ↳98 PiDP

                ↳99 UsableRulesProof (⇔)

                ↳100 PiDP

                ↳101 PiDPToQDPProof (⇐)

                ↳102 QDP

                ↳103 QDPSizeChangeProof (⇔)

                ↳104 YES

(0) Obligation:

Clauses:

shapes(Matrix, N) :- ','(varmat(Matrix, MatrixWithVars), unif_matrx(MatrixWithVars)).
varmat([], []).
varmat(.(L, Ls), .(VL, VLs)) :- ','(varmat(L, VL), varmat(Ls, VLs)).
varmat(.(black, Xs), .(black, VXs)) :- varmat(Xs, VXs).
varmat(.(white, Xs), .(w(X1), VXs)) :- varmat(Xs, VXs).
unif_matrx(.(L1, .(L2, Ls))) :- ','(unif_lines(L1, L2), unif_matrx(.(L2, Ls))).
unif_matrx(.(X2, [])).
unif_lines(.(W, .(X, L1s)), .(Y, .(Z, L2s))) :- ','(unif_pairs(.(W, .(X, .(Y, .(Z, .(W, .(Y, .(X, .(Z, .(W, .(Z, .(X, .(Y, []))))))))))))), unif_lines(.(X, L1s), .(Z, L2s))).
unif_lines(.(X3, []), .(X4, [])).
unif_pairs([]).
unif_pairs(.(A, .(B, Pairs))) :- ','(unif(A, B), unif_pairs(Pairs)).
unif(w(A), w(A)).
unif(black, black).
unif(black, w(X5)).
unif(w(X6), black).

Queries:

shapes(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: shapes(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: shapes(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(varmat(T5, X11), unif_matrx(X11))\n\nT5 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 1 | ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 2 | ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 3 | ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 1\n\nT5 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 2 | ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 3 | ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: unif_matrx([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: unif_matrx([]), SCOPE: 3, CLAUSE: 5 | unif_matrx([]), SCOPE: 3, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: unif_matrx([]), SCOPE: 3, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 3 | ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(varmat(T15, X49), ','(varmat(T16, X50), unif_matrx(.(X49, X50))))\n\nT15 is ground\nT16 is ground\nX49 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: varmat(T15, X49)\n\nT15 is ground\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(varmat(T16, X50), unif_matrx(.(T17, X50)))\n\nT16 is ground\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: varmat(T15, X49), SCOPE: 4, CLAUSE: 1 | varmat(T15, X49), SCOPE: 4, CLAUSE: 2 | varmat(T15, X49), SCOPE: 4, CLAUSE: 3 | varmat(T15, X49), SCOPE: 4, CLAUSE: 4\n\nT15 is ground\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: varmat(T15, X49), SCOPE: 4, CLAUSE: 1\n\nT15 is ground\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: varmat(T15, X49), SCOPE: 4, CLAUSE: 2 | varmat(T15, X49), SCOPE: 4, CLAUSE: 3 | varmat(T15, X49), SCOPE: 4, CLAUSE: 4\n\nT15 is ground\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: varmat(T15, X49), SCOPE: 4, CLAUSE: 2\n\nT15 is ground\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: varmat(T15, X49), SCOPE: 4, CLAUSE: 3 | varmat(T15, X49), SCOPE: 4, CLAUSE: 4\n\nT15 is ground\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(varmat(T26, X83), varmat(T27, X84))\n\nT26 is ground\nT27 is ground\nX83 is free; \nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: varmat(T26, X83)\n\nT26 is ground\nX83 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: varmat(T27, X84)\n\nT27 is ground\nX84 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: varmat(T15, X49), SCOPE: 4, CLAUSE: 3\n\nT15 is ground\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: varmat(T15, X49), SCOPE: 4, CLAUSE: 4\n\nT15 is ground\nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: varmat(T33, X103)\n\nT33 is ground\nX103 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: varmat(T36, X120)\n\nT36 is ground\nX120 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: varmat(T16, X50)\n\nT16 is ground\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: unif_matrx(.(T17, T37))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: unif_matrx(.(T17, T37)), SCOPE: 5, CLAUSE: 5 | unif_matrx(.(T17, T37)), SCOPE: 5, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: unif_matrx(.(T17, T37)), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: unif_matrx(.(T17, T37)), SCOPE: 5, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(unif_lines(T56, T57), unif_matrx(.(T57, T58)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: unif_lines(T56, T57)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: unif_matrx(.(T65, T66))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: unif_lines(T56, T57), SCOPE: 6, CLAUSE: 7 | unif_lines(T56, T57), SCOPE: 6, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: unif_lines(T56, T57), SCOPE: 6, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: unif_lines(T56, T57), SCOPE: 6, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(unif_pairs(.(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))))), unif_lines(.(T104, T107), .(T106, T108)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(unif_pairs(.(T103, T113)), unif_lines(.(T104, T107), .(T106, T108)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: unif_pairs(.(T103, T113))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: unif_lines(.(T117, T118), .(T119, T120))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: unif_pairs(.(T103, T113)), SCOPE: 7, CLAUSE: 9 | unif_pairs(.(T103, T113)), SCOPE: 7, CLAUSE: 10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: unif_pairs(.(T103, T113)), SCOPE: 7, CLAUSE: 10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(unif(T133, T134), unif_pairs(T135))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 11 | ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 12 | ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 13 | ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 12 | ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 13 | ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: unif_pairs(T141)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: unif_pairs(T141), SCOPE: 9, CLAUSE: 9 | unif_pairs(T141), SCOPE: 9, CLAUSE: 10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: unif_pairs(T141), SCOPE: 9, CLAUSE: 9\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: unif_pairs(T141), SCOPE: 9, CLAUSE: 10\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(unif(T151, T152), unif_pairs(T153))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 11 | ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 12 | ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 13 | ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 12 | ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 13 | ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: unif_pairs(T159)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 12\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 13 | ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: unif_pairs(T160)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 13\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(unif(T151, T152), unif_pairs(T153)), SCOPE: 10, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: unif_pairs(T166)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: unif_pairs(T170)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 12\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 13 | ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: unif_pairs(T171)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 13\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(unif(T133, T134), unif_pairs(T135)), SCOPE: 8, CLAUSE: 14\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: unif_pairs(T177)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: unif_pairs(T181)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: ','(varmat(T5, X11), unif_matrx(X11)), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX11 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(varmat(T204, X293), unif_matrx(.(black, X293)))\n\nT204 is ground\nX293 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: ','(varmat(T207, X310), unif_matrx(.(w(X309), X310)))\n\nT207 is ground\nX309 is free; \nX310 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 107 [arrowhead = none ];
107 -> 2;

108 [label="ONLY EVAL with clause\nshapes(X9, X10) :- ','(varmat(X9, X11), unif_matrx(X11)).\nand substitution\nT1 -> T5\nX9 -> T5\nT2 -> T6\nX10 -> T6", fontsize=14, style = filled, fillcolor = white];
2 -> 108 [arrowhead = none ];
108 -> 3;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 109 [arrowhead = none ];
109 -> 4;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 110 [arrowhead = none ];
110 -> 5;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 111 [arrowhead = none ];
111 -> 6;

112 [label="EVAL with clause\nvarmat([], []).\nand substitution\nT5 -> []\nX11 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 112 [arrowhead = none ];
112 -> 7;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 113 [arrowhead = none ];
113 -> 8;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 114 [arrowhead = none ];
114 -> 12;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 115 [arrowhead = none ];
115 -> 13;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 116 [arrowhead = none ];
116 -> 9;

117 [label="BACKTRACK\nfor clause: unif_matrx(.(L1, .(L2, Ls))) :- ','(unif_lines(L1, L2), unif_matrx(.(L2, Ls)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
9 -> 117 [arrowhead = none ];
117 -> 10;

118 [label="BACKTRACK\nfor clause: unif_matrx(.(X2, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 118 [arrowhead = none ];
118 -> 11;

119 [label="EVAL with clause\nvarmat(.(X45, X46), .(X47, X48)) :- ','(varmat(X45, X47), varmat(X46, X48)).\nand substitution\nX45 -> T15\nX46 -> T16\nT5 -> .(T15, T16)\nX47 -> X49\nX48 -> X50\nX11 -> .(X49, X50)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 119 [arrowhead = none ];
119 -> 14;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 120 [arrowhead = none ];
120 -> 15;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 121 [arrowhead = none ];
121 -> 101;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 122 [arrowhead = none ];
122 -> 102;

123 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
14 -> 123 [arrowhead = none ];
123 -> 16;

124 [label="SPLIT 2\nreplacements:\nX49 -> T17", fontsize=14, style = filled, fillcolor = violet];
14 -> 124 [arrowhead = none ];
124 -> 17;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 125 [arrowhead = none ];
125 -> 18;

126 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 126 [arrowhead = none ];
126 -> 36;

127 [label="SPLIT 2\nreplacements:\nX50 -> T37", fontsize=14, style = filled, fillcolor = violet];
17 -> 127 [arrowhead = none ];
127 -> 37;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 128 [arrowhead = none ];
128 -> 19;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 129 [arrowhead = none ];
129 -> 20;

130 [label="EVAL with clause\nvarmat([], []).\nand substitution\nT15 -> []\nX49 -> []", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 130 [arrowhead = none ];
130 -> 21;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 131 [arrowhead = none ];
131 -> 22;

132 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 132 [arrowhead = none ];
132 -> 24;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 133 [arrowhead = none ];
133 -> 25;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 134 [arrowhead = none ];
134 -> 23;

135 [label="EVAL with clause\nvarmat(.(X79, X80), .(X81, X82)) :- ','(varmat(X79, X81), varmat(X80, X82)).\nand substitution\nX79 -> T26\nX80 -> T27\nT15 -> .(T26, T27)\nX81 -> X83\nX82 -> X84\nX49 -> .(X83, X84)", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 135 [arrowhead = none ];
135 -> 26;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 136 [arrowhead = none ];
136 -> 27;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 137 [arrowhead = none ];
137 -> 30;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 138 [arrowhead = none ];
138 -> 31;

139 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
26 -> 139 [arrowhead = none ];
139 -> 28;

140 [label="SPLIT 2\nreplacements:\nX83 -> T28", fontsize=14, style = filled, fillcolor = violet];
26 -> 140 [arrowhead = none ];
140 -> 29;

141 [label="INSTANCE with matching:\nT15 -> T26\nX49 -> X83", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 141 [arrowhead = none , style=dashed];
141 -> 16[style=dashed];

142 [label="INSTANCE with matching:\nT15 -> T27\nX49 -> X84", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 142 [arrowhead = none , style=dashed];
142 -> 16[style=dashed];

143 [label="EVAL with clause\nvarmat(.(black, X101), .(black, X102)) :- varmat(X101, X102).\nand substitution\nX101 -> T33\nT15 -> .(black, T33)\nX102 -> X103\nX49 -> .(black, X103)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 143 [arrowhead = none ];
143 -> 32;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 144 [arrowhead = none ];
144 -> 33;

145 [label="EVAL with clause\nvarmat(.(white, X116), .(w(X117), X118)) :- varmat(X116, X118).\nand substitution\nX116 -> T36\nT15 -> .(white, T36)\nX117 -> X119\nX118 -> X120\nX49 -> .(w(X119), X120)", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 145 [arrowhead = none ];
145 -> 34;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 146 [arrowhead = none ];
146 -> 35;

147 [label="INSTANCE with matching:\nT15 -> T33\nX49 -> X103", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 147 [arrowhead = none , style=dashed];
147 -> 16[style=dashed];

148 [label="INSTANCE with matching:\nT15 -> T36\nX49 -> X120", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 148 [arrowhead = none , style=dashed];
148 -> 16[style=dashed];

149 [label="INSTANCE with matching:\nT15 -> T16\nX49 -> X50", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 149 [arrowhead = none , style=dashed];
149 -> 16[style=dashed];

150 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 150 [arrowhead = none ];
150 -> 38;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 151 [arrowhead = none ];
151 -> 39;

152 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 152 [arrowhead = none ];
152 -> 40;

153 [label="EVAL with clause\nunif_matrx(.(X139, .(X140, X141))) :- ','(unif_lines(X139, X140), unif_matrx(.(X140, X141))).\nand substitution\nT17 -> T56\nX139 -> T56\nX140 -> T57\nX141 -> T58\nT37 -> .(T57, T58)\nT53 -> T56\nT54 -> T57\nT55 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 153 [arrowhead = none ];
153 -> 41;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 154 [arrowhead = none ];
154 -> 42;

155 [label="EVAL with clause\nunif_matrx(.(X277, [])).\nand substitution\nT17 -> T199\nX277 -> T199\nT37 -> []", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 155 [arrowhead = none ];
155 -> 98;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 156 [arrowhead = none ];
156 -> 99;

157 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 157 [arrowhead = none ];
157 -> 43;

158 [label="SPLIT 2\nreplacements:\nT57 -> T65\nT58 -> T66", fontsize=14, style = filled, fillcolor = violet];
41 -> 158 [arrowhead = none ];
158 -> 44;

159 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 159 [arrowhead = none ];
159 -> 45;

160 [label="INSTANCE with matching:\nT17 -> T65\nT37 -> T66", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 160 [arrowhead = none , style=dashed];
160 -> 37[style=dashed];

161 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 161 [arrowhead = none ];
161 -> 46;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 162 [arrowhead = none ];
162 -> 47;

163 [label="EVAL with clause\nunif_lines(.(X181, .(X182, X183)), .(X184, .(X185, X186))) :- ','(unif_pairs(.(X181, .(X182, .(X184, .(X185, .(X181, .(X184, .(X182, .(X185, .(X181, .(X185, .(X182, .(X184, []))))))))))))), unif_lines(.(X182, X183), .(X185, X186))).\nand substitution\nX181 -> T103\nX182 -> T104\nX183 -> T107\nT56 -> .(T103, .(T104, T107))\nX184 -> T105\nX185 -> T106\nX186 -> T108\nT57 -> .(T105, .(T106, T108))\nT97 -> T103\nT98 -> T104\nT100 -> T105\nT101 -> T106\nT99 -> T107\nT102 -> T108", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 163 [arrowhead = none ];
163 -> 48;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 164 [arrowhead = none ];
164 -> 49;

165 [label="EVAL with clause\nunif_lines(.(X268, []), .(X269, [])).\nand substitution\nX268 -> T192\nT56 -> .(T192, [])\nX269 -> T193\nT57 -> .(T193, [])", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 165 [arrowhead = none ];
165 -> 95;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 166 [arrowhead = none ];
166 -> 96;

167 [label="GENERALIZATION\nT113 <-- .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, [])))))))))))", fontsize=14, style = filled, fillcolor = violet];
48 -> 167 [arrowhead = none ];
167 -> 50;

168 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 168 [arrowhead = none ];
168 -> 51;

169 [label="SPLIT 2\nreplacements:\nT104 -> T117\nT107 -> T118\nT106 -> T119\nT108 -> T120", fontsize=14, style = filled, fillcolor = violet];
50 -> 169 [arrowhead = none ];
169 -> 52;

170 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 170 [arrowhead = none ];
170 -> 53;

171 [label="INSTANCE with matching:\nT56 -> .(T117, T118)\nT57 -> .(T119, T120)", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 171 [arrowhead = none , style=dashed];
171 -> 43[style=dashed];

172 [label="BACKTRACK\nfor clause: unif_pairs([])because of non-unification", fontsize=14, style = filled, fillcolor = white];
53 -> 172 [arrowhead = none ];
172 -> 54;

173 [label="EVAL with clause\nunif_pairs(.(X208, .(X209, X210))) :- ','(unif(X208, X209), unif_pairs(X210)).\nand substitution\nT103 -> T133\nX208 -> T133\nX209 -> T134\nX210 -> T135\nT113 -> .(T134, T135)\nT130 -> T133\nT131 -> T134\nT132 -> T135", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 173 [arrowhead = none ];
173 -> 55;

174 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 174 [arrowhead = none ];
174 -> 56;

175 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 175 [arrowhead = none ];
175 -> 57;

176 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 176 [arrowhead = none ];
176 -> 58;

177 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 177 [arrowhead = none ];
177 -> 59;

178 [label="EVAL with clause\nunif(w(X218), w(X218)).\nand substitution\nX218 -> T140\nT133 -> w(T140)\nT134 -> w(T140)\nT135 -> T141", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 178 [arrowhead = none ];
178 -> 60;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 179 [arrowhead = none ];
179 -> 61;

180 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 180 [arrowhead = none ];
180 -> 85;

181 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 181 [arrowhead = none ];
181 -> 86;

182 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 182 [arrowhead = none ];
182 -> 62;

183 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 183 [arrowhead = none ];
183 -> 63;

184 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 184 [arrowhead = none ];
184 -> 64;

185 [label="EVAL with clause\nunif_pairs([]).\nand substitution\nT141 -> []", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 185 [arrowhead = none ];
185 -> 65;

186 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 186 [arrowhead = none ];
186 -> 66;

187 [label="EVAL with clause\nunif_pairs(.(X226, .(X227, X228))) :- ','(unif(X226, X227), unif_pairs(X228)).\nand substitution\nX226 -> T151\nX227 -> T152\nX228 -> T153\nT141 -> .(T151, .(T152, T153))\nT148 -> T151\nT149 -> T152\nT150 -> T153", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 187 [arrowhead = none ];
187 -> 68;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 188 [arrowhead = none ];
188 -> 69;

189 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 189 [arrowhead = none ];
189 -> 67;

190 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 190 [arrowhead = none ];
190 -> 70;

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
70 -> 191 [arrowhead = none ];
191 -> 71;

192 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
70 -> 192 [arrowhead = none ];
192 -> 72;

193 [label="EVAL with clause\nunif(w(X236), w(X236)).\nand substitution\nX236 -> T158\nT151 -> w(T158)\nT152 -> w(T158)\nT153 -> T159", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 193 [arrowhead = none ];
193 -> 73;

194 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 194 [arrowhead = none ];
194 -> 74;

195 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 195 [arrowhead = none ];
195 -> 75;

196 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 196 [arrowhead = none ];
196 -> 76;

197 [label="INSTANCE with matching:\nT141 -> T159", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 197 [arrowhead = none , style=dashed];
197 -> 60[style=dashed];

198 [label="EVAL with clause\nunif(black, black).\nand substitution\nT151 -> black\nT152 -> black\nT153 -> T160", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 198 [arrowhead = none ];
198 -> 77;

199 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 199 [arrowhead = none ];
199 -> 78;

200 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 200 [arrowhead = none ];
200 -> 79;

201 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 201 [arrowhead = none ];
201 -> 80;

202 [label="INSTANCE with matching:\nT141 -> T160", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 202 [arrowhead = none , style=dashed];
202 -> 60[style=dashed];

203 [label="EVAL with clause\nunif(black, w(X242)).\nand substitution\nT151 -> black\nX242 -> T165\nT152 -> w(T165)\nT153 -> T166", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 203 [arrowhead = none ];
203 -> 81;

204 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 204 [arrowhead = none ];
204 -> 82;

205 [label="EVAL with clause\nunif(w(X246), black).\nand substitution\nX246 -> T169\nT151 -> w(T169)\nT152 -> black\nT153 -> T170", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 205 [arrowhead = none ];
205 -> 83;

206 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 206 [arrowhead = none ];
206 -> 84;

207 [label="INSTANCE with matching:\nT141 -> T166", fontsize=14, style = filled, fillcolor = lightgrey];
81 -> 207 [arrowhead = none , style=dashed];
207 -> 60[style=dashed];

208 [label="INSTANCE with matching:\nT141 -> T170", fontsize=14, style = filled, fillcolor = lightgrey];
83 -> 208 [arrowhead = none , style=dashed];
208 -> 60[style=dashed];

209 [label="EVAL with clause\nunif(black, black).\nand substitution\nT133 -> black\nT134 -> black\nT135 -> T171", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 209 [arrowhead = none ];
209 -> 87;

210 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 210 [arrowhead = none ];
210 -> 88;

211 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 211 [arrowhead = none ];
211 -> 89;

212 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 212 [arrowhead = none ];
212 -> 90;

213 [label="INSTANCE with matching:\nT141 -> T171", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 213 [arrowhead = none , style=dashed];
213 -> 60[style=dashed];

214 [label="EVAL with clause\nunif(black, w(X252)).\nand substitution\nT133 -> black\nX252 -> T176\nT134 -> w(T176)\nT135 -> T177", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 214 [arrowhead = none ];
214 -> 91;

215 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 215 [arrowhead = none ];
215 -> 92;

216 [label="EVAL with clause\nunif(w(X256), black).\nand substitution\nX256 -> T180\nT133 -> w(T180)\nT134 -> black\nT135 -> T181", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 216 [arrowhead = none ];
216 -> 93;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 217 [arrowhead = none ];
217 -> 94;

218 [label="INSTANCE with matching:\nT141 -> T177", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 218 [arrowhead = none , style=dashed];
218 -> 60[style=dashed];

219 [label="INSTANCE with matching:\nT141 -> T181", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 219 [arrowhead = none , style=dashed];
219 -> 60[style=dashed];

220 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 220 [arrowhead = none ];
220 -> 97;

221 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
98 -> 221 [arrowhead = none ];
221 -> 100;

222 [label="EVAL with clause\nvarmat(.(black, X291), .(black, X292)) :- varmat(X291, X292).\nand substitution\nX291 -> T204\nT5 -> .(black, T204)\nX292 -> X293\nX11 -> .(black, X293)", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 222 [arrowhead = none ];
222 -> 103;

223 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 223 [arrowhead = none ];
223 -> 104;

224 [label="EVAL with clause\nvarmat(.(white, X306), .(w(X307), X308)) :- varmat(X306, X308).\nand substitution\nX306 -> T207\nT5 -> .(white, T207)\nX307 -> X309\nX308 -> X310\nX11 -> .(w(X309), X310)", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 224 [arrowhead = none ];
224 -> 105;

225 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 225 [arrowhead = none ];
225 -> 106;

226 [label="INSTANCE with matching:\nT16 -> T204\nX50 -> X293\nT17 -> black", fontsize=14, style = filled, fillcolor = lightgrey];
103 -> 226 [arrowhead = none , style=dashed];
226 -> 17[style=dashed];

227 [label="INSTANCE with matching:\nT16 -> T207\nX50 -> X310\nT17 -> w(X309)", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 227 [arrowhead = none , style=dashed];
227 -> 17[style=dashed];

}

(2) Obligation:

Clauses:

varmat16([], []).
varmat16(.(T26, T27), .(X83, X84)) :- varmat16(T26, X83).
varmat16(.(T26, T27), .(T28, X84)) :- ','(varmat16(T26, T28), varmat16(T27, X84)).
varmat16(.(black, T33), .(black, X103)) :- varmat16(T33, X103).
varmat16(.(white, T36), .(w(X119), X120)) :- varmat16(T36, X120).
unif_matrx37(T56, .(T57, T58)) :- unif_lines43(T56, T57).
unif_matrx37(T56, .(T65, T66)) :- ','(unif_lines43(T56, T65), unif_matrx37(T65, T66)).
unif_matrx37(T199, []).
unif_lines43(.(T103, .(T104, T107)), .(T105, .(T106, T108))) :- p50(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108).
unif_lines43(.(T192, []), .(T193, [])).
unif_pairs60([]).
unif_pairs60(.(w(T158), .(w(T158), T159))) :- unif_pairs60(T159).
unif_pairs60(.(black, .(black, T160))) :- unif_pairs60(T160).
unif_pairs60(.(black, .(w(T165), T166))) :- unif_pairs60(T166).
unif_pairs60(.(w(T169), .(black, T170))) :- unif_pairs60(T170).
p17(T16, X50, T17) :- varmat16(T16, X50).
p17(T16, T37, T17) :- ','(varmat16(T16, T37), unif_matrx37(T17, T37)).
p50(T103, T113, T104, T107, T106, T108) :- unif_pairs51(T103, T113).
p50(T103, T113, T117, T118, T119, T120) :- ','(unif_pairs51(T103, T113), unif_lines43(.(T117, T118), .(T119, T120))).
unif_pairs51(w(T140), .(w(T140), T141)) :- unif_pairs60(T141).
unif_pairs51(black, .(black, T171)) :- unif_pairs60(T171).
unif_pairs51(black, .(w(T176), T177)) :- unif_pairs60(T177).
unif_pairs51(w(T180), .(black, T181)) :- unif_pairs60(T181).
shapes1(.(T15, T16), T6) :- varmat16(T15, X49).
shapes1(.(T15, T16), T6) :- ','(varmat16(T15, T17), p17(T16, X50, T17)).
shapes1(.(black, T204), T6) :- p17(T204, X293, black).
shapes1(.(white, T207), T6) :- p17(T207, X310, w(X309)).

Queries:

shapes1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
shapes1_in: (b,f)
varmat16_in: (b,f)
p17_in: (b,f,f) (b,f,b)
unif_matrx37_in: (f,f) (b,f)
unif_lines43_in: (f,f) (b,f)
p50_in: (f,f,f,f,f,f) (b,f,b,b,f,f)
unif_pairs51_in: (f,f) (b,f)
unif_pairs60_in: (f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SHAPES1_IN_GA(.(T15, T16), T6) → U24_GA(T15, T16, T6, varmat16_in_ga(T15, X49))
SHAPES1_IN_GA(.(T15, T16), T6) → VARMAT16_IN_GA(T15, X49)
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → U1_GA(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → VARMAT16_IN_GA(T26, X83)
VARMAT16_IN_GA(.(T26, T27), .(T28, X84)) → U2_GA(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → U4_GA(T33, X103, varmat16_in_ga(T33, X103))
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → VARMAT16_IN_GA(T33, X103)
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → U5_GA(T36, X119, X120, varmat16_in_ga(T36, X120))
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → VARMAT16_IN_GA(T36, X120)
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_GA(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → VARMAT16_IN_GA(T27, X84)
SHAPES1_IN_GA(.(T15, T16), T6) → U25_GA(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_GA(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_GA(T15, T16, T6, p17_in_gaa(T16, X50, T17))
U25_GA(T15, T16, T6, varmat16_out_ga(T15, T17)) → P17_IN_GAA(T16, X50, T17)
P17_IN_GAA(T16, X50, T17) → U14_GAA(T16, X50, T17, varmat16_in_ga(T16, X50))
P17_IN_GAA(T16, X50, T17) → VARMAT16_IN_GA(T16, X50)
P17_IN_GAA(T16, T37, T17) → U15_GAA(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_GAA(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_GAA(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
U15_GAA(T16, T37, T17, varmat16_out_ga(T16, T37)) → UNIF_MATRX37_IN_AA(T17, T37)
UNIF_MATRX37_IN_AA(T56, .(T57, T58)) → U6_AA(T56, T57, T58, unif_lines43_in_aa(T56, T57))
UNIF_MATRX37_IN_AA(T56, .(T57, T58)) → UNIF_LINES43_IN_AA(T56, T57)
UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_AA(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_AAAAAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_AAAAAA(T103, T113, T104, T107, T106, T108) → U17_AAAAAA(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
P50_IN_AAAAAA(T103, T113, T104, T107, T106, T108) → UNIF_PAIRS51_IN_AA(T103, T113)
UNIF_PAIRS51_IN_AA(w(T140), .(w(T140), T141)) → U20_AA(T140, T141, unif_pairs60_in_a(T141))
UNIF_PAIRS51_IN_AA(w(T140), .(w(T140), T141)) → UNIF_PAIRS60_IN_A(T141)
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → U10_A(T158, T159, unif_pairs60_in_a(T159))
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → UNIF_PAIRS60_IN_A(T159)
UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → U11_A(T160, unif_pairs60_in_a(T160))
UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → UNIF_PAIRS60_IN_A(T160)
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → U12_A(T165, T166, unif_pairs60_in_a(T166))
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → UNIF_PAIRS60_IN_A(T166)
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → U13_A(T169, T170, unif_pairs60_in_a(T170))
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → UNIF_PAIRS60_IN_A(T170)
UNIF_PAIRS51_IN_AA(black, .(black, T171)) → U21_AA(T171, unif_pairs60_in_a(T171))
UNIF_PAIRS51_IN_AA(black, .(black, T171)) → UNIF_PAIRS60_IN_A(T171)
UNIF_PAIRS51_IN_AA(black, .(w(T176), T177)) → U22_AA(T176, T177, unif_pairs60_in_a(T177))
UNIF_PAIRS51_IN_AA(black, .(w(T176), T177)) → UNIF_PAIRS60_IN_A(T177)
UNIF_PAIRS51_IN_AA(w(T180), .(black, T181)) → U23_AA(T180, T181, unif_pairs60_in_a(T181))
UNIF_PAIRS51_IN_AA(w(T180), .(black, T181)) → UNIF_PAIRS60_IN_A(T181)
P50_IN_AAAAAA(T103, T113, T117, T118, T119, T120) → U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_AAAAAA(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA(.(T117, T118), .(T119, T120))
UNIF_MATRX37_IN_AA(T56, .(T65, T66)) → U7_AA(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_AA(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)
SHAPES1_IN_GA(.(black, T204), T6) → U27_GA(T204, T6, p17_in_gag(T204, X293, black))
SHAPES1_IN_GA(.(black, T204), T6) → P17_IN_GAG(T204, X293, black)
P17_IN_GAG(T16, X50, T17) → U14_GAG(T16, X50, T17, varmat16_in_ga(T16, X50))
P17_IN_GAG(T16, X50, T17) → VARMAT16_IN_GA(T16, X50)
P17_IN_GAG(T16, T37, T17) → U15_GAG(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_GAG(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_GAG(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
U15_GAG(T16, T37, T17, varmat16_out_ga(T16, T37)) → UNIF_MATRX37_IN_GA(T17, T37)
UNIF_MATRX37_IN_GA(T56, .(T57, T58)) → U6_GA(T56, T57, T58, unif_lines43_in_ga(T56, T57))
UNIF_MATRX37_IN_GA(T56, .(T57, T58)) → UNIF_LINES43_IN_GA(T56, T57)
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_GA(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_GAGGAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_GAGGAA(T103, T113, T104, T107, T106, T108) → U17_GAGGAA(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
P50_IN_GAGGAA(T103, T113, T104, T107, T106, T108) → UNIF_PAIRS51_IN_GA(T103, T113)
UNIF_PAIRS51_IN_GA(w(T140), .(w(T140), T141)) → U20_GA(T140, T141, unif_pairs60_in_a(T141))
UNIF_PAIRS51_IN_GA(w(T140), .(w(T140), T141)) → UNIF_PAIRS60_IN_A(T141)
UNIF_PAIRS51_IN_GA(black, .(black, T171)) → U21_GA(T171, unif_pairs60_in_a(T171))
UNIF_PAIRS51_IN_GA(black, .(black, T171)) → UNIF_PAIRS60_IN_A(T171)
UNIF_PAIRS51_IN_GA(black, .(w(T176), T177)) → U22_GA(T176, T177, unif_pairs60_in_a(T177))
UNIF_PAIRS51_IN_GA(black, .(w(T176), T177)) → UNIF_PAIRS60_IN_A(T177)
UNIF_PAIRS51_IN_GA(w(T180), .(black, T181)) → U23_GA(T180, T181, unif_pairs60_in_a(T181))
UNIF_PAIRS51_IN_GA(w(T180), .(black, T181)) → UNIF_PAIRS60_IN_A(T181)
P50_IN_GAGGAA(T103, T113, T117, T118, T119, T120) → U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_GAGGAA(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118), .(T119, T120))
UNIF_MATRX37_IN_GA(T56, .(T65, T66)) → U7_GA(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_GA(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_GA(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U7_GA(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)
SHAPES1_IN_GA(.(white, T207), T6) → U28_GA(T207, T6, p17_in_gag(T207, X310, w(X309)))
SHAPES1_IN_GA(.(white, T207), T6) → P17_IN_GAG(T207, X310, w(X309))

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

The argument filtering Pi contains the following mapping:
shapes1_in_ga(x1, x2) = shapes1_in_ga(x1)
.(x1, x2) = .(x1, x2)
U24_ga(x1, x2, x3, x4) = U24_ga(x1, x2, x4)
varmat16_in_ga(x1, x2) = varmat16_in_ga(x1)
[] = []
varmat16_out_ga(x1, x2) = varmat16_out_ga(x1)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x5)
black = black
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
white = white
U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4)
w(x1) = w
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5)
shapes1_out_ga(x1, x2) = shapes1_out_ga(x1)
U25_ga(x1, x2, x3, x4) = U25_ga(x1, x2, x4)
U26_ga(x1, x2, x3, x4) = U26_ga(x1, x2, x4)
p17_in_gaa(x1, x2, x3) = p17_in_gaa(x1)
U14_gaa(x1, x2, x3, x4) = U14_gaa(x1, x4)
p17_out_gaa(x1, x2, x3) = p17_out_gaa(x1)
U15_gaa(x1, x2, x3, x4) = U15_gaa(x1, x4)
U16_gaa(x1, x2, x3, x4) = U16_gaa(x1, x4)
unif_matrx37_in_aa(x1, x2) = unif_matrx37_in_aa
U6_aa(x1, x2, x3, x4) = U6_aa(x4)
unif_lines43_in_aa(x1, x2) = unif_lines43_in_aa
U9_aa(x1, x2, x3, x4, x5, x6, x7) = U9_aa(x7)
p50_in_aaaaaa(x1, x2, x3, x4, x5, x6) = p50_in_aaaaaa
U17_aaaaaa(x1, x2, x3, x4, x5, x6, x7) = U17_aaaaaa(x7)
unif_pairs51_in_aa(x1, x2) = unif_pairs51_in_aa
U20_aa(x1, x2, x3) = U20_aa(x3)
unif_pairs60_in_a(x1) = unif_pairs60_in_a
unif_pairs60_out_a(x1) = unif_pairs60_out_a(x1)
U10_a(x1, x2, x3) = U10_a(x3)
U11_a(x1, x2) = U11_a(x2)
U12_a(x1, x2, x3) = U12_a(x3)
U13_a(x1, x2, x3) = U13_a(x3)
unif_pairs51_out_aa(x1, x2) = unif_pairs51_out_aa(x1, x2)
U21_aa(x1, x2) = U21_aa(x2)
U22_aa(x1, x2, x3) = U22_aa(x3)
U23_aa(x1, x2, x3) = U23_aa(x3)
p50_out_aaaaaa(x1, x2, x3, x4, x5, x6) = p50_out_aaaaaa(x1, x2)
U18_aaaaaa(x1, x2, x3, x4, x5, x6, x7) = U18_aaaaaa(x7)
U19_aaaaaa(x1, x2, x3, x4, x5, x6, x7) = U19_aaaaaa(x1, x2, x7)
unif_lines43_out_aa(x1, x2) = unif_lines43_out_aa
unif_matrx37_out_aa(x1, x2) = unif_matrx37_out_aa
U7_aa(x1, x2, x3, x4) = U7_aa(x4)
U8_aa(x1, x2, x3, x4) = U8_aa(x4)
U27_ga(x1, x2, x3) = U27_ga(x1, x3)
p17_in_gag(x1, x2, x3) = p17_in_gag(x1, x3)
U14_gag(x1, x2, x3, x4) = U14_gag(x1, x3, x4)
p17_out_gag(x1, x2, x3) = p17_out_gag(x1, x3)
U15_gag(x1, x2, x3, x4) = U15_gag(x1, x3, x4)
U16_gag(x1, x2, x3, x4) = U16_gag(x1, x3, x4)
unif_matrx37_in_ga(x1, x2) = unif_matrx37_in_ga(x1)
U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4)
unif_lines43_in_ga(x1, x2) = unif_lines43_in_ga(x1)
U9_ga(x1, x2, x3, x4, x5, x6, x7) = U9_ga(x1, x2, x3, x7)
p50_in_gaggaa(x1, x2, x3, x4, x5, x6) = p50_in_gaggaa(x1, x3, x4)
U17_gaggaa(x1, x2, x3, x4, x5, x6, x7) = U17_gaggaa(x1, x3, x4, x7)
unif_pairs51_in_ga(x1, x2) = unif_pairs51_in_ga(x1)
U20_ga(x1, x2, x3) = U20_ga(x3)
unif_pairs51_out_ga(x1, x2) = unif_pairs51_out_ga(x1, x2)
U21_ga(x1, x2) = U21_ga(x2)
U22_ga(x1, x2, x3) = U22_ga(x3)
U23_ga(x1, x2, x3) = U23_ga(x3)
p50_out_gaggaa(x1, x2, x3, x4, x5, x6) = p50_out_gaggaa(x1, x2, x3, x4)
U18_gaggaa(x1, x2, x3, x4, x5, x6, x7) = U18_gaggaa(x1, x3, x4, x7)
U19_gaggaa(x1, x2, x3, x4, x5, x6, x7) = U19_gaggaa(x1, x2, x3, x4, x7)
unif_lines43_out_ga(x1, x2) = unif_lines43_out_ga(x1)
unif_matrx37_out_ga(x1, x2) = unif_matrx37_out_ga(x1)
U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4)
U8_ga(x1, x2, x3, x4) = U8_ga(x1, x4)
U28_ga(x1, x2, x3) = U28_ga(x1, x3)
SHAPES1_IN_GA(x1, x2) = SHAPES1_IN_GA(x1)
U24_GA(x1, x2, x3, x4) = U24_GA(x1, x2, x4)
VARMAT16_IN_GA(x1, x2) = VARMAT16_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x5)
U4_GA(x1, x2, x3) = U4_GA(x1, x3)
U5_GA(x1, x2, x3, x4) = U5_GA(x1, x4)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5)
U25_GA(x1, x2, x3, x4) = U25_GA(x1, x2, x4)
U26_GA(x1, x2, x3, x4) = U26_GA(x1, x2, x4)
P17_IN_GAA(x1, x2, x3) = P17_IN_GAA(x1)
U14_GAA(x1, x2, x3, x4) = U14_GAA(x1, x4)
U15_GAA(x1, x2, x3, x4) = U15_GAA(x1, x4)
U16_GAA(x1, x2, x3, x4) = U16_GAA(x1, x4)
UNIF_MATRX37_IN_AA(x1, x2) = UNIF_MATRX37_IN_AA
U6_AA(x1, x2, x3, x4) = U6_AA(x4)
UNIF_LINES43_IN_AA(x1, x2) = UNIF_LINES43_IN_AA
U9_AA(x1, x2, x3, x4, x5, x6, x7) = U9_AA(x7)
P50_IN_AAAAAA(x1, x2, x3, x4, x5, x6) = P50_IN_AAAAAA
U17_AAAAAA(x1, x2, x3, x4, x5, x6, x7) = U17_AAAAAA(x7)
UNIF_PAIRS51_IN_AA(x1, x2) = UNIF_PAIRS51_IN_AA
U20_AA(x1, x2, x3) = U20_AA(x3)
UNIF_PAIRS60_IN_A(x1) = UNIF_PAIRS60_IN_A
U10_A(x1, x2, x3) = U10_A(x3)
U11_A(x1, x2) = U11_A(x2)
U12_A(x1, x2, x3) = U12_A(x3)
U13_A(x1, x2, x3) = U13_A(x3)
U21_AA(x1, x2) = U21_AA(x2)
U22_AA(x1, x2, x3) = U22_AA(x3)
U23_AA(x1, x2, x3) = U23_AA(x3)
U18_AAAAAA(x1, x2, x3, x4, x5, x6, x7) = U18_AAAAAA(x7)
U19_AAAAAA(x1, x2, x3, x4, x5, x6, x7) = U19_AAAAAA(x1, x2, x7)
U7_AA(x1, x2, x3, x4) = U7_AA(x4)
U8_AA(x1, x2, x3, x4) = U8_AA(x4)
U27_GA(x1, x2, x3) = U27_GA(x1, x3)
P17_IN_GAG(x1, x2, x3) = P17_IN_GAG(x1, x3)
U14_GAG(x1, x2, x3, x4) = U14_GAG(x1, x3, x4)
U15_GAG(x1, x2, x3, x4) = U15_GAG(x1, x3, x4)
U16_GAG(x1, x2, x3, x4) = U16_GAG(x1, x3, x4)
UNIF_MATRX37_IN_GA(x1, x2) = UNIF_MATRX37_IN_GA(x1)
U6_GA(x1, x2, x3, x4) = U6_GA(x1, x4)
UNIF_LINES43_IN_GA(x1, x2) = UNIF_LINES43_IN_GA(x1)
U9_GA(x1, x2, x3, x4, x5, x6, x7) = U9_GA(x1, x2, x3, x7)
P50_IN_GAGGAA(x1, x2, x3, x4, x5, x6) = P50_IN_GAGGAA(x1, x3, x4)
U17_GAGGAA(x1, x2, x3, x4, x5, x6, x7) = U17_GAGGAA(x1, x3, x4, x7)
UNIF_PAIRS51_IN_GA(x1, x2) = UNIF_PAIRS51_IN_GA(x1)
U20_GA(x1, x2, x3) = U20_GA(x3)
U21_GA(x1, x2) = U21_GA(x2)
U22_GA(x1, x2, x3) = U22_GA(x3)
U23_GA(x1, x2, x3) = U23_GA(x3)
U18_GAGGAA(x1, x2, x3, x4, x5, x6, x7) = U18_GAGGAA(x1, x3, x4, x7)
U19_GAGGAA(x1, x2, x3, x4, x5, x6, x7) = U19_GAGGAA(x1, x2, x3, x4, x7)
U7_GA(x1, x2, x3, x4) = U7_GA(x1, x4)
U8_GA(x1, x2, x3, x4) = U8_GA(x1, x4)
U28_GA(x1, x2, x3) = U28_GA(x1, x3)

We have to consider all (P,R,Pi)-chains

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SHAPES1_IN_GA(.(T15, T16), T6) → U24_GA(T15, T16, T6, varmat16_in_ga(T15, X49))
SHAPES1_IN_GA(.(T15, T16), T6) → VARMAT16_IN_GA(T15, X49)
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → U1_GA(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → VARMAT16_IN_GA(T26, X83)
VARMAT16_IN_GA(.(T26, T27), .(T28, X84)) → U2_GA(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → U4_GA(T33, X103, varmat16_in_ga(T33, X103))
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → VARMAT16_IN_GA(T33, X103)
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → U5_GA(T36, X119, X120, varmat16_in_ga(T36, X120))
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → VARMAT16_IN_GA(T36, X120)
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_GA(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → VARMAT16_IN_GA(T27, X84)
SHAPES1_IN_GA(.(T15, T16), T6) → U25_GA(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_GA(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_GA(T15, T16, T6, p17_in_gaa(T16, X50, T17))
U25_GA(T15, T16, T6, varmat16_out_ga(T15, T17)) → P17_IN_GAA(T16, X50, T17)
P17_IN_GAA(T16, X50, T17) → U14_GAA(T16, X50, T17, varmat16_in_ga(T16, X50))
P17_IN_GAA(T16, X50, T17) → VARMAT16_IN_GA(T16, X50)
P17_IN_GAA(T16, T37, T17) → U15_GAA(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_GAA(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_GAA(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
U15_GAA(T16, T37, T17, varmat16_out_ga(T16, T37)) → UNIF_MATRX37_IN_AA(T17, T37)
UNIF_MATRX37_IN_AA(T56, .(T57, T58)) → U6_AA(T56, T57, T58, unif_lines43_in_aa(T56, T57))
UNIF_MATRX37_IN_AA(T56, .(T57, T58)) → UNIF_LINES43_IN_AA(T56, T57)
UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_AA(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_AAAAAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_AAAAAA(T103, T113, T104, T107, T106, T108) → U17_AAAAAA(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
P50_IN_AAAAAA(T103, T113, T104, T107, T106, T108) → UNIF_PAIRS51_IN_AA(T103, T113)
UNIF_PAIRS51_IN_AA(w(T140), .(w(T140), T141)) → U20_AA(T140, T141, unif_pairs60_in_a(T141))
UNIF_PAIRS51_IN_AA(w(T140), .(w(T140), T141)) → UNIF_PAIRS60_IN_A(T141)
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → U10_A(T158, T159, unif_pairs60_in_a(T159))
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → UNIF_PAIRS60_IN_A(T159)
UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → U11_A(T160, unif_pairs60_in_a(T160))
UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → UNIF_PAIRS60_IN_A(T160)
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → U12_A(T165, T166, unif_pairs60_in_a(T166))
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → UNIF_PAIRS60_IN_A(T166)
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → U13_A(T169, T170, unif_pairs60_in_a(T170))
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → UNIF_PAIRS60_IN_A(T170)
UNIF_PAIRS51_IN_AA(black, .(black, T171)) → U21_AA(T171, unif_pairs60_in_a(T171))
UNIF_PAIRS51_IN_AA(black, .(black, T171)) → UNIF_PAIRS60_IN_A(T171)
UNIF_PAIRS51_IN_AA(black, .(w(T176), T177)) → U22_AA(T176, T177, unif_pairs60_in_a(T177))
UNIF_PAIRS51_IN_AA(black, .(w(T176), T177)) → UNIF_PAIRS60_IN_A(T177)
UNIF_PAIRS51_IN_AA(w(T180), .(black, T181)) → U23_AA(T180, T181, unif_pairs60_in_a(T181))
UNIF_PAIRS51_IN_AA(w(T180), .(black, T181)) → UNIF_PAIRS60_IN_A(T181)
P50_IN_AAAAAA(T103, T113, T117, T118, T119, T120) → U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_AAAAAA(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA(.(T117, T118), .(T119, T120))
UNIF_MATRX37_IN_AA(T56, .(T65, T66)) → U7_AA(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_AA(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)
SHAPES1_IN_GA(.(black, T204), T6) → U27_GA(T204, T6, p17_in_gag(T204, X293, black))
SHAPES1_IN_GA(.(black, T204), T6) → P17_IN_GAG(T204, X293, black)
P17_IN_GAG(T16, X50, T17) → U14_GAG(T16, X50, T17, varmat16_in_ga(T16, X50))
P17_IN_GAG(T16, X50, T17) → VARMAT16_IN_GA(T16, X50)
P17_IN_GAG(T16, T37, T17) → U15_GAG(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_GAG(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_GAG(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
U15_GAG(T16, T37, T17, varmat16_out_ga(T16, T37)) → UNIF_MATRX37_IN_GA(T17, T37)
UNIF_MATRX37_IN_GA(T56, .(T57, T58)) → U6_GA(T56, T57, T58, unif_lines43_in_ga(T56, T57))
UNIF_MATRX37_IN_GA(T56, .(T57, T58)) → UNIF_LINES43_IN_GA(T56, T57)
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_GA(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_GAGGAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_GAGGAA(T103, T113, T104, T107, T106, T108) → U17_GAGGAA(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
P50_IN_GAGGAA(T103, T113, T104, T107, T106, T108) → UNIF_PAIRS51_IN_GA(T103, T113)
UNIF_PAIRS51_IN_GA(w(T140), .(w(T140), T141)) → U20_GA(T140, T141, unif_pairs60_in_a(T141))
UNIF_PAIRS51_IN_GA(w(T140), .(w(T140), T141)) → UNIF_PAIRS60_IN_A(T141)
UNIF_PAIRS51_IN_GA(black, .(black, T171)) → U21_GA(T171, unif_pairs60_in_a(T171))
UNIF_PAIRS51_IN_GA(black, .(black, T171)) → UNIF_PAIRS60_IN_A(T171)
UNIF_PAIRS51_IN_GA(black, .(w(T176), T177)) → U22_GA(T176, T177, unif_pairs60_in_a(T177))
UNIF_PAIRS51_IN_GA(black, .(w(T176), T177)) → UNIF_PAIRS60_IN_A(T177)
UNIF_PAIRS51_IN_GA(w(T180), .(black, T181)) → U23_GA(T180, T181, unif_pairs60_in_a(T181))
UNIF_PAIRS51_IN_GA(w(T180), .(black, T181)) → UNIF_PAIRS60_IN_A(T181)
P50_IN_GAGGAA(T103, T113, T117, T118, T119, T120) → U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_GAGGAA(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118), .(T119, T120))
UNIF_MATRX37_IN_GA(T56, .(T65, T66)) → U7_GA(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_GA(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_GA(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U7_GA(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)
SHAPES1_IN_GA(.(white, T207), T6) → U28_GA(T207, T6, p17_in_gag(T207, X310, w(X309)))
SHAPES1_IN_GA(.(white, T207), T6) → P17_IN_GAG(T207, X310, w(X309))

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 59 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → UNIF_PAIRS60_IN_A(T160)
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → UNIF_PAIRS60_IN_A(T159)
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → UNIF_PAIRS60_IN_A(T166)
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → UNIF_PAIRS60_IN_A(T170)

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → UNIF_PAIRS60_IN_A(T160)
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → UNIF_PAIRS60_IN_A(T159)
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → UNIF_PAIRS60_IN_A(T166)
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → UNIF_PAIRS60_IN_A(T170)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
black = black
w(x1) = w
UNIF_PAIRS60_IN_A(x1) = UNIF_PAIRS60_IN_A

We have to consider all (P,R,Pi)-chains

(12) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_PAIRS60_IN_A → UNIF_PAIRS60_IN_A

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = UNIF_PAIRS60_IN_A evaluates to t =UNIF_PAIRS60_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from UNIF_PAIRS60_IN_A to UNIF_PAIRS60_IN_A.

(15) NO

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P50_IN_GAGGAA(T103, T113, T117, T118, T119, T120) → U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118), .(T119, T120))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_GAGGAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P50_IN_GAGGAA(T103, T113, T117, T118, T119, T120) → U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118), .(T119, T120))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_GAGGAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)

The TRS R consists of the following rules:

unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
black = black
w(x1) = w
unif_pairs60_in_a(x1) = unif_pairs60_in_a
unif_pairs60_out_a(x1) = unif_pairs60_out_a(x1)
U10_a(x1, x2, x3) = U10_a(x3)
U11_a(x1, x2) = U11_a(x2)
U12_a(x1, x2, x3) = U12_a(x3)
U13_a(x1, x2, x3) = U13_a(x3)
unif_pairs51_in_ga(x1, x2) = unif_pairs51_in_ga(x1)
U20_ga(x1, x2, x3) = U20_ga(x3)
unif_pairs51_out_ga(x1, x2) = unif_pairs51_out_ga(x1, x2)
U21_ga(x1, x2) = U21_ga(x2)
U22_ga(x1, x2, x3) = U22_ga(x3)
U23_ga(x1, x2, x3) = U23_ga(x3)
UNIF_LINES43_IN_GA(x1, x2) = UNIF_LINES43_IN_GA(x1)
P50_IN_GAGGAA(x1, x2, x3, x4, x5, x6) = P50_IN_GAGGAA(x1, x3, x4)
U18_GAGGAA(x1, x2, x3, x4, x5, x6, x7) = U18_GAGGAA(x1, x3, x4, x7)

We have to consider all (P,R,Pi)-chains

(19) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P50_IN_GAGGAA(T103, T117, T118) → U18_GAGGAA(T103, T117, T118, unif_pairs51_in_ga(T103))
U18_GAGGAA(T103, T117, T118, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107))) → P50_IN_GAGGAA(T103, T104, T107)

The TRS R consists of the following rules:

unif_pairs51_in_ga(w) → U20_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(black) → U21_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(black) → U22_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(w) → U23_ga(unif_pairs60_in_a)
U20_ga(unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w, .(w, T141))
U21_ga(unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
U22_ga(unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w, T177))
U23_ga(unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_pairs51_in_ga(x0)
U20_ga(x0)
U21_ga(x0)
U22_ga(x0)
U23_ga(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNIF_LINES43_IN_GA(.(T103, .(T104, T107))) → P50_IN_GAGGAA(T103, T104, T107)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂
POL(P50_IN_GAGGAA(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U10_a(x₁)) = 0
POL(U11_a(x₁)) = 0
POL(U12_a(x₁)) = 0
POL(U13_a(x₁)) = 0
POL(U18_GAGGAA(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃
POL(U20_ga(x₁)) = 0
POL(U21_ga(x₁)) = 0
POL(U22_ga(x₁)) = 0
POL(U23_ga(x₁)) = 0
POL(UNIF_LINES43_IN_GA(x₁)) = x₁
POL([]) = 0
POL(black) = 0
POL(unif_pairs51_in_ga(x₁)) = 0
POL(unif_pairs51_out_ga(x₁, x₂)) = 0
POL(unif_pairs60_in_a) = 0
POL(unif_pairs60_out_a(x₁)) = 0
POL(w) = 0

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P50_IN_GAGGAA(T103, T117, T118) → U18_GAGGAA(T103, T117, T118, unif_pairs51_in_ga(T103))
U18_GAGGAA(T103, T117, T118, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118))

The TRS R consists of the following rules:

unif_pairs51_in_ga(w) → U20_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(black) → U21_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(black) → U22_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(w) → U23_ga(unif_pairs60_in_a)
U20_ga(unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w, .(w, T141))
U21_ga(unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
U22_ga(unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w, T177))
U23_ga(unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_pairs51_in_ga(x0)
U20_ga(x0)
U21_ga(x0)
U22_ga(x0)
U23_ga(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(24) TRUE

(25) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_AAAAAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_AAAAAA(T103, T113, T117, T118, T119, T120) → U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA(.(T117, T118), .(T119, T120))

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(26) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(27) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_AAAAAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_AAAAAA(T103, T113, T117, T118, T119, T120) → U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA(.(T117, T118), .(T119, T120))

The TRS R consists of the following rules:

unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
black = black
w(x1) = w
unif_pairs51_in_aa(x1, x2) = unif_pairs51_in_aa
U20_aa(x1, x2, x3) = U20_aa(x3)
unif_pairs60_in_a(x1) = unif_pairs60_in_a
unif_pairs60_out_a(x1) = unif_pairs60_out_a(x1)
U10_a(x1, x2, x3) = U10_a(x3)
U11_a(x1, x2) = U11_a(x2)
U12_a(x1, x2, x3) = U12_a(x3)
U13_a(x1, x2, x3) = U13_a(x3)
unif_pairs51_out_aa(x1, x2) = unif_pairs51_out_aa(x1, x2)
U21_aa(x1, x2) = U21_aa(x2)
U22_aa(x1, x2, x3) = U22_aa(x3)
U23_aa(x1, x2, x3) = U23_aa(x3)
UNIF_LINES43_IN_AA(x1, x2) = UNIF_LINES43_IN_AA
P50_IN_AAAAAA(x1, x2, x3, x4, x5, x6) = P50_IN_AAAAAA
U18_AAAAAA(x1, x2, x3, x4, x5, x6, x7) = U18_AAAAAA(x7)

We have to consider all (P,R,Pi)-chains

(28) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(29) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA → P50_IN_AAAAAA
P50_IN_AAAAAA → U18_AAAAAA(unif_pairs51_in_aa)
U18_AAAAAA(unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA

The TRS R consists of the following rules:

unif_pairs51_in_aa → U20_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U21_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U22_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U23_aa(unif_pairs60_in_a)
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_pairs51_in_aa
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P50_IN_AAAAAA → U18_AAAAAA(unif_pairs51_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U21_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U22_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U23_aa(unif_pairs60_in_a))

(31) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA → P50_IN_AAAAAA
U18_AAAAAA(unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA
P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U21_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U22_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U23_aa(unif_pairs60_in_a))

The TRS R consists of the following rules:

unif_pairs51_in_aa → U20_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U21_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U22_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U23_aa(unif_pairs60_in_a)
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_pairs51_in_aa
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(32) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(33) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA → P50_IN_AAAAAA
U18_AAAAAA(unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA
P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U21_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U22_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U23_aa(unif_pairs60_in_a))

The TRS R consists of the following rules:

unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))

The set Q consists of the following terms:

unif_pairs51_in_aa
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(34) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

unif_pairs51_in_aa

(35) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA → P50_IN_AAAAAA
U18_AAAAAA(unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA
P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U21_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U22_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U23_aa(unif_pairs60_in_a))

The TRS R consists of the following rules:

unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))

The set Q consists of the following terms:

U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(36) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U18_AAAAAA(U20_aa(unif_pairs60_in_a)) evaluates to t =U18_AAAAAA(U20_aa(unif_pairs60_in_a))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U18_AAAAAA(U20_aa(unif_pairs60_in_a)) → U18_AAAAAA(U20_aa(unif_pairs60_out_a([])))
with rule unif_pairs60_in_a → unif_pairs60_out_a([]) at position [0,0] and matcher [ ]

U18_AAAAAA(U20_aa(unif_pairs60_out_a([]))) → U18_AAAAAA(unif_pairs51_out_aa(w, .(w, [])))
with rule U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141)) at position [0] and matcher [T141 / []]

U18_AAAAAA(unif_pairs51_out_aa(w, .(w, []))) → UNIF_LINES43_IN_AA
with rule U18_AAAAAA(unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA at position [] and matcher [T103 / w, T113 / .(w, [])]

UNIF_LINES43_IN_AA → P50_IN_AAAAAA
with rule UNIF_LINES43_IN_AA → P50_IN_AAAAAA at position [] and matcher [ ]

P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))
with rule P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(37) NO

(38) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_MATRX37_IN_AA(T56, .(T65, T66)) → U7_AA(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(39) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(40) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_MATRX37_IN_AA(T56, .(T65, T66)) → U7_AA(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)

The TRS R consists of the following rules:

unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
black = black
w(x1) = w
unif_lines43_in_aa(x1, x2) = unif_lines43_in_aa
U9_aa(x1, x2, x3, x4, x5, x6, x7) = U9_aa(x7)
p50_in_aaaaaa(x1, x2, x3, x4, x5, x6) = p50_in_aaaaaa
U17_aaaaaa(x1, x2, x3, x4, x5, x6, x7) = U17_aaaaaa(x7)
unif_pairs51_in_aa(x1, x2) = unif_pairs51_in_aa
U20_aa(x1, x2, x3) = U20_aa(x3)
unif_pairs60_in_a(x1) = unif_pairs60_in_a
unif_pairs60_out_a(x1) = unif_pairs60_out_a(x1)
U10_a(x1, x2, x3) = U10_a(x3)
U11_a(x1, x2) = U11_a(x2)
U12_a(x1, x2, x3) = U12_a(x3)
U13_a(x1, x2, x3) = U13_a(x3)
unif_pairs51_out_aa(x1, x2) = unif_pairs51_out_aa(x1, x2)
U21_aa(x1, x2) = U21_aa(x2)
U22_aa(x1, x2, x3) = U22_aa(x3)
U23_aa(x1, x2, x3) = U23_aa(x3)
p50_out_aaaaaa(x1, x2, x3, x4, x5, x6) = p50_out_aaaaaa(x1, x2)
U18_aaaaaa(x1, x2, x3, x4, x5, x6, x7) = U18_aaaaaa(x7)
U19_aaaaaa(x1, x2, x3, x4, x5, x6, x7) = U19_aaaaaa(x1, x2, x7)
unif_lines43_out_aa(x1, x2) = unif_lines43_out_aa
UNIF_MATRX37_IN_AA(x1, x2) = UNIF_MATRX37_IN_AA
U7_AA(x1, x2, x3, x4) = U7_AA(x4)

We have to consider all (P,R,Pi)-chains

(41) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(42) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_in_aa)
U7_AA(unif_lines43_out_aa) → UNIF_MATRX37_IN_AA

The TRS R consists of the following rules:

unif_lines43_in_aa → U9_aa(p50_in_aaaaaa)
unif_lines43_in_aa → unif_lines43_out_aa
U9_aa(p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))))) → unif_lines43_out_aa
p50_in_aaaaaa → U17_aaaaaa(unif_pairs51_in_aa)
p50_in_aaaaaa → U18_aaaaaa(unif_pairs51_in_aa)
U17_aaaaaa(unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113)
U18_aaaaaa(unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, unif_lines43_in_aa)
unif_pairs51_in_aa → U20_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U21_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U22_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U23_aa(unif_pairs60_in_a)
U19_aaaaaa(T103, T113, unif_lines43_out_aa) → p50_out_aaaaaa(T103, T113)
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_lines43_in_aa
U9_aa(x0)
p50_in_aaaaaa
U17_aaaaaa(x0)
U18_aaaaaa(x0)
unif_pairs51_in_aa
U19_aaaaaa(x0, x1, x2)
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(43) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_in_aa) at position [0] we obtained the following new rules [LPAR04]:

UNIF_MATRX37_IN_AA → U7_AA(U9_aa(p50_in_aaaaaa))
UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_out_aa)

(44) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U7_AA(unif_lines43_out_aa) → UNIF_MATRX37_IN_AA
UNIF_MATRX37_IN_AA → U7_AA(U9_aa(p50_in_aaaaaa))
UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_out_aa)

The TRS R consists of the following rules:

unif_lines43_in_aa → U9_aa(p50_in_aaaaaa)
unif_lines43_in_aa → unif_lines43_out_aa
U9_aa(p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))))) → unif_lines43_out_aa
p50_in_aaaaaa → U17_aaaaaa(unif_pairs51_in_aa)
p50_in_aaaaaa → U18_aaaaaa(unif_pairs51_in_aa)
U17_aaaaaa(unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113)
U18_aaaaaa(unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, unif_lines43_in_aa)
unif_pairs51_in_aa → U20_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U21_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U22_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U23_aa(unif_pairs60_in_a)
U19_aaaaaa(T103, T113, unif_lines43_out_aa) → p50_out_aaaaaa(T103, T113)
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_lines43_in_aa
U9_aa(x0)
p50_in_aaaaaa
U17_aaaaaa(x0)
U18_aaaaaa(x0)
unif_pairs51_in_aa
U19_aaaaaa(x0, x1, x2)
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(45) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the right:

s = U7_AA(unif_lines43_out_aa) evaluates to t =U7_AA(unif_lines43_out_aa)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U7_AA(unif_lines43_out_aa) → UNIF_MATRX37_IN_AA
with rule U7_AA(unif_lines43_out_aa) → UNIF_MATRX37_IN_AA and matcher [ ].

UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_out_aa)
with rule UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_out_aa) at position [] and matcher [ ]

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(46) NO

(47) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

VARMAT16_IN_GA(.(T26, T27), .(T28, X84)) → U2_GA(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → VARMAT16_IN_GA(T27, X84)
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → VARMAT16_IN_GA(T26, X83)
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → VARMAT16_IN_GA(T33, X103)
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → VARMAT16_IN_GA(T36, X120)

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(48) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(49) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

VARMAT16_IN_GA(.(T26, T27), .(T28, X84)) → U2_GA(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → VARMAT16_IN_GA(T27, X84)
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → VARMAT16_IN_GA(T26, X83)
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → VARMAT16_IN_GA(T33, X103)
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → VARMAT16_IN_GA(T36, X120)

The TRS R consists of the following rules:

varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
varmat16_in_ga(x1, x2) = varmat16_in_ga(x1)
[] = []
varmat16_out_ga(x1, x2) = varmat16_out_ga(x1)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x5)
black = black
U4_ga(x1, x2, x3) = U4_ga(x1, x3)
white = white
U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4)
w(x1) = w
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5)
VARMAT16_IN_GA(x1, x2) = VARMAT16_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x5)

We have to consider all (P,R,Pi)-chains

(50) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(51) Obligation:

Q DP problem:
The TRS P consists of the following rules:

VARMAT16_IN_GA(.(T26, T27)) → U2_GA(T26, T27, varmat16_in_ga(T26))
U2_GA(T26, T27, varmat16_out_ga(T26)) → VARMAT16_IN_GA(T27)
VARMAT16_IN_GA(.(T26, T27)) → VARMAT16_IN_GA(T26)
VARMAT16_IN_GA(.(black, T33)) → VARMAT16_IN_GA(T33)
VARMAT16_IN_GA(.(white, T36)) → VARMAT16_IN_GA(T36)

The TRS R consists of the following rules:

varmat16_in_ga([]) → varmat16_out_ga([])
varmat16_in_ga(.(T26, T27)) → U1_ga(T26, T27, varmat16_in_ga(T26))
varmat16_in_ga(.(T26, T27)) → U2_ga(T26, T27, varmat16_in_ga(T26))
varmat16_in_ga(.(black, T33)) → U4_ga(T33, varmat16_in_ga(T33))
varmat16_in_ga(.(white, T36)) → U5_ga(T36, varmat16_in_ga(T36))
U1_ga(T26, T27, varmat16_out_ga(T26)) → varmat16_out_ga(.(T26, T27))
U2_ga(T26, T27, varmat16_out_ga(T26)) → U3_ga(T26, T27, varmat16_in_ga(T27))
U4_ga(T33, varmat16_out_ga(T33)) → varmat16_out_ga(.(black, T33))
U5_ga(T36, varmat16_out_ga(T36)) → varmat16_out_ga(.(white, T36))
U3_ga(T26, T27, varmat16_out_ga(T27)) → varmat16_out_ga(.(T26, T27))

The set Q consists of the following terms:

varmat16_in_ga(x0)
U1_ga(x0, x1, x2)
U2_ga(x0, x1, x2)
U4_ga(x0, x1)
U5_ga(x0, x1)
U3_ga(x0, x1, x2)

We have to consider all (P,Q,R)-chains.

(52) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

U2_GA(T26, T27, varmat16_out_ga(T26)) → VARMAT16_IN_GA(T27)
The graph contains the following edges 2 >= 1
VARMAT16_IN_GA(.(T26, T27)) → U2_GA(T26, T27, varmat16_in_ga(T26))
The graph contains the following edges 1 > 1, 1 > 2
VARMAT16_IN_GA(.(T26, T27)) → VARMAT16_IN_GA(T26)
The graph contains the following edges 1 > 1
VARMAT16_IN_GA(.(black, T33)) → VARMAT16_IN_GA(T33)
The graph contains the following edges 1 > 1
VARMAT16_IN_GA(.(white, T36)) → VARMAT16_IN_GA(T36)
The graph contains the following edges 1 > 1

(53) YES

(54) PrologToPiTRSProof (SOUND transformation)

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(55) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(56) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SHAPES1_IN_GA(.(T15, T16), T6) → U24_GA(T15, T16, T6, varmat16_in_ga(T15, X49))
SHAPES1_IN_GA(.(T15, T16), T6) → VARMAT16_IN_GA(T15, X49)
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → U1_GA(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → VARMAT16_IN_GA(T26, X83)
VARMAT16_IN_GA(.(T26, T27), .(T28, X84)) → U2_GA(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → U4_GA(T33, X103, varmat16_in_ga(T33, X103))
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → VARMAT16_IN_GA(T33, X103)
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → U5_GA(T36, X119, X120, varmat16_in_ga(T36, X120))
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → VARMAT16_IN_GA(T36, X120)
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_GA(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → VARMAT16_IN_GA(T27, X84)
SHAPES1_IN_GA(.(T15, T16), T6) → U25_GA(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_GA(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_GA(T15, T16, T6, p17_in_gaa(T16, X50, T17))
U25_GA(T15, T16, T6, varmat16_out_ga(T15, T17)) → P17_IN_GAA(T16, X50, T17)
P17_IN_GAA(T16, X50, T17) → U14_GAA(T16, X50, T17, varmat16_in_ga(T16, X50))
P17_IN_GAA(T16, X50, T17) → VARMAT16_IN_GA(T16, X50)
P17_IN_GAA(T16, T37, T17) → U15_GAA(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_GAA(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_GAA(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
U15_GAA(T16, T37, T17, varmat16_out_ga(T16, T37)) → UNIF_MATRX37_IN_AA(T17, T37)
UNIF_MATRX37_IN_AA(T56, .(T57, T58)) → U6_AA(T56, T57, T58, unif_lines43_in_aa(T56, T57))
UNIF_MATRX37_IN_AA(T56, .(T57, T58)) → UNIF_LINES43_IN_AA(T56, T57)
UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_AA(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_AAAAAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_AAAAAA(T103, T113, T104, T107, T106, T108) → U17_AAAAAA(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
P50_IN_AAAAAA(T103, T113, T104, T107, T106, T108) → UNIF_PAIRS51_IN_AA(T103, T113)
UNIF_PAIRS51_IN_AA(w(T140), .(w(T140), T141)) → U20_AA(T140, T141, unif_pairs60_in_a(T141))
UNIF_PAIRS51_IN_AA(w(T140), .(w(T140), T141)) → UNIF_PAIRS60_IN_A(T141)
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → U10_A(T158, T159, unif_pairs60_in_a(T159))
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → UNIF_PAIRS60_IN_A(T159)
UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → U11_A(T160, unif_pairs60_in_a(T160))
UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → UNIF_PAIRS60_IN_A(T160)
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → U12_A(T165, T166, unif_pairs60_in_a(T166))
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → UNIF_PAIRS60_IN_A(T166)
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → U13_A(T169, T170, unif_pairs60_in_a(T170))
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → UNIF_PAIRS60_IN_A(T170)
UNIF_PAIRS51_IN_AA(black, .(black, T171)) → U21_AA(T171, unif_pairs60_in_a(T171))
UNIF_PAIRS51_IN_AA(black, .(black, T171)) → UNIF_PAIRS60_IN_A(T171)
UNIF_PAIRS51_IN_AA(black, .(w(T176), T177)) → U22_AA(T176, T177, unif_pairs60_in_a(T177))
UNIF_PAIRS51_IN_AA(black, .(w(T176), T177)) → UNIF_PAIRS60_IN_A(T177)
UNIF_PAIRS51_IN_AA(w(T180), .(black, T181)) → U23_AA(T180, T181, unif_pairs60_in_a(T181))
UNIF_PAIRS51_IN_AA(w(T180), .(black, T181)) → UNIF_PAIRS60_IN_A(T181)
P50_IN_AAAAAA(T103, T113, T117, T118, T119, T120) → U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_AAAAAA(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA(.(T117, T118), .(T119, T120))
UNIF_MATRX37_IN_AA(T56, .(T65, T66)) → U7_AA(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_AA(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)
SHAPES1_IN_GA(.(black, T204), T6) → U27_GA(T204, T6, p17_in_gag(T204, X293, black))
SHAPES1_IN_GA(.(black, T204), T6) → P17_IN_GAG(T204, X293, black)
P17_IN_GAG(T16, X50, T17) → U14_GAG(T16, X50, T17, varmat16_in_ga(T16, X50))
P17_IN_GAG(T16, X50, T17) → VARMAT16_IN_GA(T16, X50)
P17_IN_GAG(T16, T37, T17) → U15_GAG(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_GAG(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_GAG(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
U15_GAG(T16, T37, T17, varmat16_out_ga(T16, T37)) → UNIF_MATRX37_IN_GA(T17, T37)
UNIF_MATRX37_IN_GA(T56, .(T57, T58)) → U6_GA(T56, T57, T58, unif_lines43_in_ga(T56, T57))
UNIF_MATRX37_IN_GA(T56, .(T57, T58)) → UNIF_LINES43_IN_GA(T56, T57)
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_GA(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_GAGGAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_GAGGAA(T103, T113, T104, T107, T106, T108) → U17_GAGGAA(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
P50_IN_GAGGAA(T103, T113, T104, T107, T106, T108) → UNIF_PAIRS51_IN_GA(T103, T113)
UNIF_PAIRS51_IN_GA(w(T140), .(w(T140), T141)) → U20_GA(T140, T141, unif_pairs60_in_a(T141))
UNIF_PAIRS51_IN_GA(w(T140), .(w(T140), T141)) → UNIF_PAIRS60_IN_A(T141)
UNIF_PAIRS51_IN_GA(black, .(black, T171)) → U21_GA(T171, unif_pairs60_in_a(T171))
UNIF_PAIRS51_IN_GA(black, .(black, T171)) → UNIF_PAIRS60_IN_A(T171)
UNIF_PAIRS51_IN_GA(black, .(w(T176), T177)) → U22_GA(T176, T177, unif_pairs60_in_a(T177))
UNIF_PAIRS51_IN_GA(black, .(w(T176), T177)) → UNIF_PAIRS60_IN_A(T177)
UNIF_PAIRS51_IN_GA(w(T180), .(black, T181)) → U23_GA(T180, T181, unif_pairs60_in_a(T181))
UNIF_PAIRS51_IN_GA(w(T180), .(black, T181)) → UNIF_PAIRS60_IN_A(T181)
P50_IN_GAGGAA(T103, T113, T117, T118, T119, T120) → U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_GAGGAA(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118), .(T119, T120))
UNIF_MATRX37_IN_GA(T56, .(T65, T66)) → U7_GA(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_GA(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_GA(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U7_GA(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)
SHAPES1_IN_GA(.(white, T207), T6) → U28_GA(T207, T6, p17_in_gag(T207, X310, w(X309)))
SHAPES1_IN_GA(.(white, T207), T6) → P17_IN_GAG(T207, X310, w(X309))

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

The argument filtering Pi contains the following mapping:
shapes1_in_ga(x1, x2) = shapes1_in_ga(x1)
.(x1, x2) = .(x1, x2)
U24_ga(x1, x2, x3, x4) = U24_ga(x4)
varmat16_in_ga(x1, x2) = varmat16_in_ga(x1)
[] = []
varmat16_out_ga(x1, x2) = varmat16_out_ga
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x2, x5)
black = black
U4_ga(x1, x2, x3) = U4_ga(x3)
white = white
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
w(x1) = w
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5)
shapes1_out_ga(x1, x2) = shapes1_out_ga
U25_ga(x1, x2, x3, x4) = U25_ga(x2, x4)
U26_ga(x1, x2, x3, x4) = U26_ga(x4)
p17_in_gaa(x1, x2, x3) = p17_in_gaa(x1)
U14_gaa(x1, x2, x3, x4) = U14_gaa(x4)
p17_out_gaa(x1, x2, x3) = p17_out_gaa
U15_gaa(x1, x2, x3, x4) = U15_gaa(x4)
U16_gaa(x1, x2, x3, x4) = U16_gaa(x4)
unif_matrx37_in_aa(x1, x2) = unif_matrx37_in_aa
U6_aa(x1, x2, x3, x4) = U6_aa(x4)
unif_lines43_in_aa(x1, x2) = unif_lines43_in_aa
U9_aa(x1, x2, x3, x4, x5, x6, x7) = U9_aa(x7)
p50_in_aaaaaa(x1, x2, x3, x4, x5, x6) = p50_in_aaaaaa
U17_aaaaaa(x1, x2, x3, x4, x5, x6, x7) = U17_aaaaaa(x7)
unif_pairs51_in_aa(x1, x2) = unif_pairs51_in_aa
U20_aa(x1, x2, x3) = U20_aa(x3)
unif_pairs60_in_a(x1) = unif_pairs60_in_a
unif_pairs60_out_a(x1) = unif_pairs60_out_a(x1)
U10_a(x1, x2, x3) = U10_a(x3)
U11_a(x1, x2) = U11_a(x2)
U12_a(x1, x2, x3) = U12_a(x3)
U13_a(x1, x2, x3) = U13_a(x3)
unif_pairs51_out_aa(x1, x2) = unif_pairs51_out_aa(x1, x2)
U21_aa(x1, x2) = U21_aa(x2)
U22_aa(x1, x2, x3) = U22_aa(x3)
U23_aa(x1, x2, x3) = U23_aa(x3)
p50_out_aaaaaa(x1, x2, x3, x4, x5, x6) = p50_out_aaaaaa(x1, x2)
U18_aaaaaa(x1, x2, x3, x4, x5, x6, x7) = U18_aaaaaa(x7)
U19_aaaaaa(x1, x2, x3, x4, x5, x6, x7) = U19_aaaaaa(x1, x2, x7)
unif_lines43_out_aa(x1, x2) = unif_lines43_out_aa
unif_matrx37_out_aa(x1, x2) = unif_matrx37_out_aa
U7_aa(x1, x2, x3, x4) = U7_aa(x4)
U8_aa(x1, x2, x3, x4) = U8_aa(x4)
U27_ga(x1, x2, x3) = U27_ga(x3)
p17_in_gag(x1, x2, x3) = p17_in_gag(x1, x3)
U14_gag(x1, x2, x3, x4) = U14_gag(x4)
p17_out_gag(x1, x2, x3) = p17_out_gag
U15_gag(x1, x2, x3, x4) = U15_gag(x3, x4)
U16_gag(x1, x2, x3, x4) = U16_gag(x4)
unif_matrx37_in_ga(x1, x2) = unif_matrx37_in_ga(x1)
U6_ga(x1, x2, x3, x4) = U6_ga(x4)
unif_lines43_in_ga(x1, x2) = unif_lines43_in_ga(x1)
U9_ga(x1, x2, x3, x4, x5, x6, x7) = U9_ga(x7)
p50_in_gaggaa(x1, x2, x3, x4, x5, x6) = p50_in_gaggaa(x1, x3, x4)
U17_gaggaa(x1, x2, x3, x4, x5, x6, x7) = U17_gaggaa(x7)
unif_pairs51_in_ga(x1, x2) = unif_pairs51_in_ga(x1)
U20_ga(x1, x2, x3) = U20_ga(x3)
unif_pairs51_out_ga(x1, x2) = unif_pairs51_out_ga(x2)
U21_ga(x1, x2) = U21_ga(x2)
U22_ga(x1, x2, x3) = U22_ga(x3)
U23_ga(x1, x2, x3) = U23_ga(x3)
p50_out_gaggaa(x1, x2, x3, x4, x5, x6) = p50_out_gaggaa(x2)
U18_gaggaa(x1, x2, x3, x4, x5, x6, x7) = U18_gaggaa(x3, x4, x7)
U19_gaggaa(x1, x2, x3, x4, x5, x6, x7) = U19_gaggaa(x2, x7)
unif_lines43_out_ga(x1, x2) = unif_lines43_out_ga
unif_matrx37_out_ga(x1, x2) = unif_matrx37_out_ga
U7_ga(x1, x2, x3, x4) = U7_ga(x4)
U8_ga(x1, x2, x3, x4) = U8_ga(x4)
U28_ga(x1, x2, x3) = U28_ga(x3)
SHAPES1_IN_GA(x1, x2) = SHAPES1_IN_GA(x1)
U24_GA(x1, x2, x3, x4) = U24_GA(x4)
VARMAT16_IN_GA(x1, x2) = VARMAT16_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x2, x5)
U4_GA(x1, x2, x3) = U4_GA(x3)
U5_GA(x1, x2, x3, x4) = U5_GA(x4)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5)
U25_GA(x1, x2, x3, x4) = U25_GA(x2, x4)
U26_GA(x1, x2, x3, x4) = U26_GA(x4)
P17_IN_GAA(x1, x2, x3) = P17_IN_GAA(x1)
U14_GAA(x1, x2, x3, x4) = U14_GAA(x4)
U15_GAA(x1, x2, x3, x4) = U15_GAA(x4)
U16_GAA(x1, x2, x3, x4) = U16_GAA(x4)
UNIF_MATRX37_IN_AA(x1, x2) = UNIF_MATRX37_IN_AA
U6_AA(x1, x2, x3, x4) = U6_AA(x4)
UNIF_LINES43_IN_AA(x1, x2) = UNIF_LINES43_IN_AA
U9_AA(x1, x2, x3, x4, x5, x6, x7) = U9_AA(x7)
P50_IN_AAAAAA(x1, x2, x3, x4, x5, x6) = P50_IN_AAAAAA
U17_AAAAAA(x1, x2, x3, x4, x5, x6, x7) = U17_AAAAAA(x7)
UNIF_PAIRS51_IN_AA(x1, x2) = UNIF_PAIRS51_IN_AA
U20_AA(x1, x2, x3) = U20_AA(x3)
UNIF_PAIRS60_IN_A(x1) = UNIF_PAIRS60_IN_A
U10_A(x1, x2, x3) = U10_A(x3)
U11_A(x1, x2) = U11_A(x2)
U12_A(x1, x2, x3) = U12_A(x3)
U13_A(x1, x2, x3) = U13_A(x3)
U21_AA(x1, x2) = U21_AA(x2)
U22_AA(x1, x2, x3) = U22_AA(x3)
U23_AA(x1, x2, x3) = U23_AA(x3)
U18_AAAAAA(x1, x2, x3, x4, x5, x6, x7) = U18_AAAAAA(x7)
U19_AAAAAA(x1, x2, x3, x4, x5, x6, x7) = U19_AAAAAA(x1, x2, x7)
U7_AA(x1, x2, x3, x4) = U7_AA(x4)
U8_AA(x1, x2, x3, x4) = U8_AA(x4)
U27_GA(x1, x2, x3) = U27_GA(x3)
P17_IN_GAG(x1, x2, x3) = P17_IN_GAG(x1, x3)
U14_GAG(x1, x2, x3, x4) = U14_GAG(x4)
U15_GAG(x1, x2, x3, x4) = U15_GAG(x3, x4)
U16_GAG(x1, x2, x3, x4) = U16_GAG(x4)
UNIF_MATRX37_IN_GA(x1, x2) = UNIF_MATRX37_IN_GA(x1)
U6_GA(x1, x2, x3, x4) = U6_GA(x4)
UNIF_LINES43_IN_GA(x1, x2) = UNIF_LINES43_IN_GA(x1)
U9_GA(x1, x2, x3, x4, x5, x6, x7) = U9_GA(x7)
P50_IN_GAGGAA(x1, x2, x3, x4, x5, x6) = P50_IN_GAGGAA(x1, x3, x4)
U17_GAGGAA(x1, x2, x3, x4, x5, x6, x7) = U17_GAGGAA(x7)
UNIF_PAIRS51_IN_GA(x1, x2) = UNIF_PAIRS51_IN_GA(x1)
U20_GA(x1, x2, x3) = U20_GA(x3)
U21_GA(x1, x2) = U21_GA(x2)
U22_GA(x1, x2, x3) = U22_GA(x3)
U23_GA(x1, x2, x3) = U23_GA(x3)
U18_GAGGAA(x1, x2, x3, x4, x5, x6, x7) = U18_GAGGAA(x3, x4, x7)
U19_GAGGAA(x1, x2, x3, x4, x5, x6, x7) = U19_GAGGAA(x2, x7)
U7_GA(x1, x2, x3, x4) = U7_GA(x4)
U8_GA(x1, x2, x3, x4) = U8_GA(x4)
U28_GA(x1, x2, x3) = U28_GA(x3)

We have to consider all (P,R,Pi)-chains

(57) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SHAPES1_IN_GA(.(T15, T16), T6) → U24_GA(T15, T16, T6, varmat16_in_ga(T15, X49))
SHAPES1_IN_GA(.(T15, T16), T6) → VARMAT16_IN_GA(T15, X49)
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → U1_GA(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → VARMAT16_IN_GA(T26, X83)
VARMAT16_IN_GA(.(T26, T27), .(T28, X84)) → U2_GA(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → U4_GA(T33, X103, varmat16_in_ga(T33, X103))
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → VARMAT16_IN_GA(T33, X103)
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → U5_GA(T36, X119, X120, varmat16_in_ga(T36, X120))
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → VARMAT16_IN_GA(T36, X120)
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_GA(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → VARMAT16_IN_GA(T27, X84)
SHAPES1_IN_GA(.(T15, T16), T6) → U25_GA(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_GA(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_GA(T15, T16, T6, p17_in_gaa(T16, X50, T17))
U25_GA(T15, T16, T6, varmat16_out_ga(T15, T17)) → P17_IN_GAA(T16, X50, T17)
P17_IN_GAA(T16, X50, T17) → U14_GAA(T16, X50, T17, varmat16_in_ga(T16, X50))
P17_IN_GAA(T16, X50, T17) → VARMAT16_IN_GA(T16, X50)
P17_IN_GAA(T16, T37, T17) → U15_GAA(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_GAA(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_GAA(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
U15_GAA(T16, T37, T17, varmat16_out_ga(T16, T37)) → UNIF_MATRX37_IN_AA(T17, T37)
UNIF_MATRX37_IN_AA(T56, .(T57, T58)) → U6_AA(T56, T57, T58, unif_lines43_in_aa(T56, T57))
UNIF_MATRX37_IN_AA(T56, .(T57, T58)) → UNIF_LINES43_IN_AA(T56, T57)
UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_AA(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_AAAAAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_AAAAAA(T103, T113, T104, T107, T106, T108) → U17_AAAAAA(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
P50_IN_AAAAAA(T103, T113, T104, T107, T106, T108) → UNIF_PAIRS51_IN_AA(T103, T113)
UNIF_PAIRS51_IN_AA(w(T140), .(w(T140), T141)) → U20_AA(T140, T141, unif_pairs60_in_a(T141))
UNIF_PAIRS51_IN_AA(w(T140), .(w(T140), T141)) → UNIF_PAIRS60_IN_A(T141)
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → U10_A(T158, T159, unif_pairs60_in_a(T159))
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → UNIF_PAIRS60_IN_A(T159)
UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → U11_A(T160, unif_pairs60_in_a(T160))
UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → UNIF_PAIRS60_IN_A(T160)
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → U12_A(T165, T166, unif_pairs60_in_a(T166))
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → UNIF_PAIRS60_IN_A(T166)
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → U13_A(T169, T170, unif_pairs60_in_a(T170))
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → UNIF_PAIRS60_IN_A(T170)
UNIF_PAIRS51_IN_AA(black, .(black, T171)) → U21_AA(T171, unif_pairs60_in_a(T171))
UNIF_PAIRS51_IN_AA(black, .(black, T171)) → UNIF_PAIRS60_IN_A(T171)
UNIF_PAIRS51_IN_AA(black, .(w(T176), T177)) → U22_AA(T176, T177, unif_pairs60_in_a(T177))
UNIF_PAIRS51_IN_AA(black, .(w(T176), T177)) → UNIF_PAIRS60_IN_A(T177)
UNIF_PAIRS51_IN_AA(w(T180), .(black, T181)) → U23_AA(T180, T181, unif_pairs60_in_a(T181))
UNIF_PAIRS51_IN_AA(w(T180), .(black, T181)) → UNIF_PAIRS60_IN_A(T181)
P50_IN_AAAAAA(T103, T113, T117, T118, T119, T120) → U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_AAAAAA(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA(.(T117, T118), .(T119, T120))
UNIF_MATRX37_IN_AA(T56, .(T65, T66)) → U7_AA(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_AA(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)
SHAPES1_IN_GA(.(black, T204), T6) → U27_GA(T204, T6, p17_in_gag(T204, X293, black))
SHAPES1_IN_GA(.(black, T204), T6) → P17_IN_GAG(T204, X293, black)
P17_IN_GAG(T16, X50, T17) → U14_GAG(T16, X50, T17, varmat16_in_ga(T16, X50))
P17_IN_GAG(T16, X50, T17) → VARMAT16_IN_GA(T16, X50)
P17_IN_GAG(T16, T37, T17) → U15_GAG(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_GAG(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_GAG(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
U15_GAG(T16, T37, T17, varmat16_out_ga(T16, T37)) → UNIF_MATRX37_IN_GA(T17, T37)
UNIF_MATRX37_IN_GA(T56, .(T57, T58)) → U6_GA(T56, T57, T58, unif_lines43_in_ga(T56, T57))
UNIF_MATRX37_IN_GA(T56, .(T57, T58)) → UNIF_LINES43_IN_GA(T56, T57)
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_GA(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_GAGGAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_GAGGAA(T103, T113, T104, T107, T106, T108) → U17_GAGGAA(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
P50_IN_GAGGAA(T103, T113, T104, T107, T106, T108) → UNIF_PAIRS51_IN_GA(T103, T113)
UNIF_PAIRS51_IN_GA(w(T140), .(w(T140), T141)) → U20_GA(T140, T141, unif_pairs60_in_a(T141))
UNIF_PAIRS51_IN_GA(w(T140), .(w(T140), T141)) → UNIF_PAIRS60_IN_A(T141)
UNIF_PAIRS51_IN_GA(black, .(black, T171)) → U21_GA(T171, unif_pairs60_in_a(T171))
UNIF_PAIRS51_IN_GA(black, .(black, T171)) → UNIF_PAIRS60_IN_A(T171)
UNIF_PAIRS51_IN_GA(black, .(w(T176), T177)) → U22_GA(T176, T177, unif_pairs60_in_a(T177))
UNIF_PAIRS51_IN_GA(black, .(w(T176), T177)) → UNIF_PAIRS60_IN_A(T177)
UNIF_PAIRS51_IN_GA(w(T180), .(black, T181)) → U23_GA(T180, T181, unif_pairs60_in_a(T181))
UNIF_PAIRS51_IN_GA(w(T180), .(black, T181)) → UNIF_PAIRS60_IN_A(T181)
P50_IN_GAGGAA(T103, T113, T117, T118, T119, T120) → U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_GAGGAA(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118), .(T119, T120))
UNIF_MATRX37_IN_GA(T56, .(T65, T66)) → U7_GA(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_GA(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_GA(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U7_GA(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)
SHAPES1_IN_GA(.(white, T207), T6) → U28_GA(T207, T6, p17_in_gag(T207, X310, w(X309)))
SHAPES1_IN_GA(.(white, T207), T6) → P17_IN_GAG(T207, X310, w(X309))

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(58) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 59 less nodes.

(59) Complex Obligation (AND)

(60) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → UNIF_PAIRS60_IN_A(T160)
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → UNIF_PAIRS60_IN_A(T159)
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → UNIF_PAIRS60_IN_A(T166)
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → UNIF_PAIRS60_IN_A(T170)

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(61) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(62) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_PAIRS60_IN_A(.(black, .(black, T160))) → UNIF_PAIRS60_IN_A(T160)
UNIF_PAIRS60_IN_A(.(w(T158), .(w(T158), T159))) → UNIF_PAIRS60_IN_A(T159)
UNIF_PAIRS60_IN_A(.(black, .(w(T165), T166))) → UNIF_PAIRS60_IN_A(T166)
UNIF_PAIRS60_IN_A(.(w(T169), .(black, T170))) → UNIF_PAIRS60_IN_A(T170)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
black = black
w(x1) = w
UNIF_PAIRS60_IN_A(x1) = UNIF_PAIRS60_IN_A

We have to consider all (P,R,Pi)-chains

(63) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(64) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_PAIRS60_IN_A → UNIF_PAIRS60_IN_A

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(65) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from UNIF_PAIRS60_IN_A to UNIF_PAIRS60_IN_A.

(66) NO

(67) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P50_IN_GAGGAA(T103, T113, T117, T118, T119, T120) → U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118), .(T119, T120))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_GAGGAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(68) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(69) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P50_IN_GAGGAA(T103, T113, T117, T118, T119, T120) → U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_GAGGAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → UNIF_LINES43_IN_GA(.(T117, T118), .(T119, T120))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_GAGGAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)

The TRS R consists of the following rules:

unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
black = black
w(x1) = w
unif_pairs60_in_a(x1) = unif_pairs60_in_a
unif_pairs60_out_a(x1) = unif_pairs60_out_a(x1)
U10_a(x1, x2, x3) = U10_a(x3)
U11_a(x1, x2) = U11_a(x2)
U12_a(x1, x2, x3) = U12_a(x3)
U13_a(x1, x2, x3) = U13_a(x3)
unif_pairs51_in_ga(x1, x2) = unif_pairs51_in_ga(x1)
U20_ga(x1, x2, x3) = U20_ga(x3)
unif_pairs51_out_ga(x1, x2) = unif_pairs51_out_ga(x2)
U21_ga(x1, x2) = U21_ga(x2)
U22_ga(x1, x2, x3) = U22_ga(x3)
U23_ga(x1, x2, x3) = U23_ga(x3)
UNIF_LINES43_IN_GA(x1, x2) = UNIF_LINES43_IN_GA(x1)
P50_IN_GAGGAA(x1, x2, x3, x4, x5, x6) = P50_IN_GAGGAA(x1, x3, x4)
U18_GAGGAA(x1, x2, x3, x4, x5, x6, x7) = U18_GAGGAA(x3, x4, x7)

We have to consider all (P,R,Pi)-chains

(70) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(71) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P50_IN_GAGGAA(T103, T117, T118) → U18_GAGGAA(T117, T118, unif_pairs51_in_ga(T103))
U18_GAGGAA(T117, T118, unif_pairs51_out_ga(T113)) → UNIF_LINES43_IN_GA(.(T117, T118))
UNIF_LINES43_IN_GA(.(T103, .(T104, T107))) → P50_IN_GAGGAA(T103, T104, T107)

The TRS R consists of the following rules:

unif_pairs51_in_ga(w) → U20_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(black) → U21_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(black) → U22_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(w) → U23_ga(unif_pairs60_in_a)
U20_ga(unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(.(w, T141))
U21_ga(unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(.(black, T171))
U22_ga(unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(.(w, T177))
U23_ga(unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(.(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_pairs51_in_ga(x0)
U20_ga(x0)
U21_ga(x0)
U22_ga(x0)
U23_ga(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNIF_LINES43_IN_GA(.(T103, .(T104, T107))) → P50_IN_GAGGAA(T103, T104, T107)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂
POL(P50_IN_GAGGAA(x₁, x₂, x₃)) = 1 + x₃
POL(U10_a(x₁)) = 0
POL(U11_a(x₁)) = 0
POL(U12_a(x₁)) = 0
POL(U13_a(x₁)) = 0
POL(U18_GAGGAA(x₁, x₂, x₃)) = 1 + x₂
POL(U20_ga(x₁)) = 0
POL(U21_ga(x₁)) = 0
POL(U22_ga(x₁)) = 0
POL(U23_ga(x₁)) = 0
POL(UNIF_LINES43_IN_GA(x₁)) = x₁
POL([]) = 0
POL(black) = 0
POL(unif_pairs51_in_ga(x₁)) = 0
POL(unif_pairs51_out_ga(x₁)) = 0
POL(unif_pairs60_in_a) = 0
POL(unif_pairs60_out_a(x₁)) = 0
POL(w) = 0

The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P50_IN_GAGGAA(T103, T117, T118) → U18_GAGGAA(T117, T118, unif_pairs51_in_ga(T103))
U18_GAGGAA(T117, T118, unif_pairs51_out_ga(T113)) → UNIF_LINES43_IN_GA(.(T117, T118))

The TRS R consists of the following rules:

unif_pairs51_in_ga(w) → U20_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(black) → U21_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(black) → U22_ga(unif_pairs60_in_a)
unif_pairs51_in_ga(w) → U23_ga(unif_pairs60_in_a)
U20_ga(unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(.(w, T141))
U21_ga(unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(.(black, T171))
U22_ga(unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(.(w, T177))
U23_ga(unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(.(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_pairs51_in_ga(x0)
U20_ga(x0)
U21_ga(x0)
U22_ga(x0)
U23_ga(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(74) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(75) TRUE

(76) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_AAAAAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_AAAAAA(T103, T113, T117, T118, T119, T120) → U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA(.(T117, T118), .(T119, T120))

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(77) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(78) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → P50_IN_AAAAAA(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)
P50_IN_AAAAAA(T103, T113, T117, T118, T119, T120) → U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_AAAAAA(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA(.(T117, T118), .(T119, T120))

The TRS R consists of the following rules:

unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))

(79) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(80) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA → P50_IN_AAAAAA
P50_IN_AAAAAA → U18_AAAAAA(unif_pairs51_in_aa)
U18_AAAAAA(unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA

The TRS R consists of the following rules:

unif_pairs51_in_aa → U20_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U21_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U22_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U23_aa(unif_pairs60_in_a)
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_pairs51_in_aa
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(81) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P50_IN_AAAAAA → U18_AAAAAA(unif_pairs51_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U21_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U22_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U23_aa(unif_pairs60_in_a))

(82) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA → P50_IN_AAAAAA
U18_AAAAAA(unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA
P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U21_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U22_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U23_aa(unif_pairs60_in_a))

The TRS R consists of the following rules:

unif_pairs51_in_aa → U20_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U21_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U22_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U23_aa(unif_pairs60_in_a)
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_pairs51_in_aa
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(83) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(84) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA → P50_IN_AAAAAA
U18_AAAAAA(unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA
P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U21_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U22_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U23_aa(unif_pairs60_in_a))

The TRS R consists of the following rules:

unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))

The set Q consists of the following terms:

unif_pairs51_in_aa
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(85) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

unif_pairs51_in_aa

(86) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_LINES43_IN_AA → P50_IN_AAAAAA
U18_AAAAAA(unif_pairs51_out_aa(T103, T113)) → UNIF_LINES43_IN_AA
P50_IN_AAAAAA → U18_AAAAAA(U20_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U21_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U22_aa(unif_pairs60_in_a))
P50_IN_AAAAAA → U18_AAAAAA(U23_aa(unif_pairs60_in_a))

The TRS R consists of the following rules:

unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))

The set Q consists of the following terms:

U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(87) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

(88) NO

(89) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_MATRX37_IN_AA(T56, .(T65, T66)) → U7_AA(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(90) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(91) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

UNIF_MATRX37_IN_AA(T56, .(T65, T66)) → U7_AA(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_AA(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → UNIF_MATRX37_IN_AA(T65, T66)

The TRS R consists of the following rules:

unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))

(92) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(93) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_in_aa)
U7_AA(unif_lines43_out_aa) → UNIF_MATRX37_IN_AA

The TRS R consists of the following rules:

unif_lines43_in_aa → U9_aa(p50_in_aaaaaa)
unif_lines43_in_aa → unif_lines43_out_aa
U9_aa(p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))))) → unif_lines43_out_aa
p50_in_aaaaaa → U17_aaaaaa(unif_pairs51_in_aa)
p50_in_aaaaaa → U18_aaaaaa(unif_pairs51_in_aa)
U17_aaaaaa(unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113)
U18_aaaaaa(unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, unif_lines43_in_aa)
unif_pairs51_in_aa → U20_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U21_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U22_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U23_aa(unif_pairs60_in_a)
U19_aaaaaa(T103, T113, unif_lines43_out_aa) → p50_out_aaaaaa(T103, T113)
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_lines43_in_aa
U9_aa(x0)
p50_in_aaaaaa
U17_aaaaaa(x0)
U18_aaaaaa(x0)
unif_pairs51_in_aa
U19_aaaaaa(x0, x1, x2)
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(94) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_in_aa) at position [0] we obtained the following new rules [LPAR04]:

UNIF_MATRX37_IN_AA → U7_AA(U9_aa(p50_in_aaaaaa))
UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_out_aa)

(95) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U7_AA(unif_lines43_out_aa) → UNIF_MATRX37_IN_AA
UNIF_MATRX37_IN_AA → U7_AA(U9_aa(p50_in_aaaaaa))
UNIF_MATRX37_IN_AA → U7_AA(unif_lines43_out_aa)

The TRS R consists of the following rules:

unif_lines43_in_aa → U9_aa(p50_in_aaaaaa)
unif_lines43_in_aa → unif_lines43_out_aa
U9_aa(p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))))) → unif_lines43_out_aa
p50_in_aaaaaa → U17_aaaaaa(unif_pairs51_in_aa)
p50_in_aaaaaa → U18_aaaaaa(unif_pairs51_in_aa)
U17_aaaaaa(unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113)
U18_aaaaaa(unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, unif_lines43_in_aa)
unif_pairs51_in_aa → U20_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U21_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U22_aa(unif_pairs60_in_a)
unif_pairs51_in_aa → U23_aa(unif_pairs60_in_a)
U19_aaaaaa(T103, T113, unif_lines43_out_aa) → p50_out_aaaaaa(T103, T113)
U20_aa(unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w, .(w, T141))
U21_aa(unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
U22_aa(unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w, T177))
U23_aa(unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w, .(black, T181))
unif_pairs60_in_a → unif_pairs60_out_a([])
unif_pairs60_in_a → U10_a(unif_pairs60_in_a)
unif_pairs60_in_a → U11_a(unif_pairs60_in_a)
unif_pairs60_in_a → U12_a(unif_pairs60_in_a)
unif_pairs60_in_a → U13_a(unif_pairs60_in_a)
U10_a(unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w, .(w, T159)))
U11_a(unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U12_a(unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w, T166)))
U13_a(unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w, .(black, T170)))

The set Q consists of the following terms:

unif_lines43_in_aa
U9_aa(x0)
p50_in_aaaaaa
U17_aaaaaa(x0)
U18_aaaaaa(x0)
unif_pairs51_in_aa
U19_aaaaaa(x0, x1, x2)
U20_aa(x0)
U21_aa(x0)
U22_aa(x0)
U23_aa(x0)
unif_pairs60_in_a
U10_a(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(96) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

(97) NO

(98) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

VARMAT16_IN_GA(.(T26, T27), .(T28, X84)) → U2_GA(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → VARMAT16_IN_GA(T27, X84)
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → VARMAT16_IN_GA(T26, X83)
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → VARMAT16_IN_GA(T33, X103)
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → VARMAT16_IN_GA(T36, X120)

The TRS R consists of the following rules:

shapes1_in_ga(.(T15, T16), T6) → U24_ga(T15, T16, T6, varmat16_in_ga(T15, X49))
varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U24_ga(T15, T16, T6, varmat16_out_ga(T15, X49)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(T15, T16), T6) → U25_ga(T15, T16, T6, varmat16_in_ga(T15, T17))
U25_ga(T15, T16, T6, varmat16_out_ga(T15, T17)) → U26_ga(T15, T16, T6, p17_in_gaa(T16, X50, T17))
p17_in_gaa(T16, X50, T17) → U14_gaa(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gaa(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gaa(T16, X50, T17)
p17_in_gaa(T16, T37, T17) → U15_gaa(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gaa(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gaa(T16, T37, T17, unif_matrx37_in_aa(T17, T37))
unif_matrx37_in_aa(T56, .(T57, T58)) → U6_aa(T56, T57, T58, unif_lines43_in_aa(T56, T57))
unif_lines43_in_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_aa(T103, T104, T107, T105, T106, T108, p50_in_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_aaaaaa(T103, T113, T104, T107, T106, T108) → U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_aa(T103, T113))
unif_pairs51_in_aa(w(T140), .(w(T140), T141)) → U20_aa(T140, T141, unif_pairs60_in_a(T141))
unif_pairs60_in_a([]) → unif_pairs60_out_a([])
unif_pairs60_in_a(.(w(T158), .(w(T158), T159))) → U10_a(T158, T159, unif_pairs60_in_a(T159))
unif_pairs60_in_a(.(black, .(black, T160))) → U11_a(T160, unif_pairs60_in_a(T160))
unif_pairs60_in_a(.(black, .(w(T165), T166))) → U12_a(T165, T166, unif_pairs60_in_a(T166))
unif_pairs60_in_a(.(w(T169), .(black, T170))) → U13_a(T169, T170, unif_pairs60_in_a(T170))
U13_a(T169, T170, unif_pairs60_out_a(T170)) → unif_pairs60_out_a(.(w(T169), .(black, T170)))
U12_a(T165, T166, unif_pairs60_out_a(T166)) → unif_pairs60_out_a(.(black, .(w(T165), T166)))
U11_a(T160, unif_pairs60_out_a(T160)) → unif_pairs60_out_a(.(black, .(black, T160)))
U10_a(T158, T159, unif_pairs60_out_a(T159)) → unif_pairs60_out_a(.(w(T158), .(w(T158), T159)))
U20_aa(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_aa(w(T140), .(w(T140), T141))
unif_pairs51_in_aa(black, .(black, T171)) → U21_aa(T171, unif_pairs60_in_a(T171))
U21_aa(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_aa(black, .(black, T171))
unif_pairs51_in_aa(black, .(w(T176), T177)) → U22_aa(T176, T177, unif_pairs60_in_a(T177))
U22_aa(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_aa(black, .(w(T176), T177))
unif_pairs51_in_aa(w(T180), .(black, T181)) → U23_aa(T180, T181, unif_pairs60_in_a(T181))
U23_aa(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_aa(w(T180), .(black, T181))
U17_aaaaaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_aa(T103, T113)) → p50_out_aaaaaa(T103, T113, T104, T107, T106, T108)
p50_in_aaaaaa(T103, T113, T117, T118, T119, T120) → U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_aa(T103, T113))
U18_aaaaaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_aa(T103, T113)) → U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_aa(.(T117, T118), .(T119, T120)))
unif_lines43_in_aa(.(T192, []), .(T193, [])) → unif_lines43_out_aa(.(T192, []), .(T193, []))
U19_aaaaaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_aa(.(T117, T118), .(T119, T120))) → p50_out_aaaaaa(T103, T113, T117, T118, T119, T120)
U9_aa(T103, T104, T107, T105, T106, T108, p50_out_aaaaaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_aa(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_aa(T56, T57, T58, unif_lines43_out_aa(T56, T57)) → unif_matrx37_out_aa(T56, .(T57, T58))
unif_matrx37_in_aa(T56, .(T65, T66)) → U7_aa(T56, T65, T66, unif_lines43_in_aa(T56, T65))
U7_aa(T56, T65, T66, unif_lines43_out_aa(T56, T65)) → U8_aa(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
unif_matrx37_in_aa(T199, []) → unif_matrx37_out_aa(T199, [])
U8_aa(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_aa(T56, .(T65, T66))
U16_gaa(T16, T37, T17, unif_matrx37_out_aa(T17, T37)) → p17_out_gaa(T16, T37, T17)
U26_ga(T15, T16, T6, p17_out_gaa(T16, X50, T17)) → shapes1_out_ga(.(T15, T16), T6)
shapes1_in_ga(.(black, T204), T6) → U27_ga(T204, T6, p17_in_gag(T204, X293, black))
p17_in_gag(T16, X50, T17) → U14_gag(T16, X50, T17, varmat16_in_ga(T16, X50))
U14_gag(T16, X50, T17, varmat16_out_ga(T16, X50)) → p17_out_gag(T16, X50, T17)
p17_in_gag(T16, T37, T17) → U15_gag(T16, T37, T17, varmat16_in_ga(T16, T37))
U15_gag(T16, T37, T17, varmat16_out_ga(T16, T37)) → U16_gag(T16, T37, T17, unif_matrx37_in_ga(T17, T37))
unif_matrx37_in_ga(T56, .(T57, T58)) → U6_ga(T56, T57, T58, unif_lines43_in_ga(T56, T57))
unif_lines43_in_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108))) → U9_ga(T103, T104, T107, T105, T106, T108, p50_in_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108))
p50_in_gaggaa(T103, T113, T104, T107, T106, T108) → U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_in_ga(T103, T113))
unif_pairs51_in_ga(w(T140), .(w(T140), T141)) → U20_ga(T140, T141, unif_pairs60_in_a(T141))
U20_ga(T140, T141, unif_pairs60_out_a(T141)) → unif_pairs51_out_ga(w(T140), .(w(T140), T141))
unif_pairs51_in_ga(black, .(black, T171)) → U21_ga(T171, unif_pairs60_in_a(T171))
U21_ga(T171, unif_pairs60_out_a(T171)) → unif_pairs51_out_ga(black, .(black, T171))
unif_pairs51_in_ga(black, .(w(T176), T177)) → U22_ga(T176, T177, unif_pairs60_in_a(T177))
U22_ga(T176, T177, unif_pairs60_out_a(T177)) → unif_pairs51_out_ga(black, .(w(T176), T177))
unif_pairs51_in_ga(w(T180), .(black, T181)) → U23_ga(T180, T181, unif_pairs60_in_a(T181))
U23_ga(T180, T181, unif_pairs60_out_a(T181)) → unif_pairs51_out_ga(w(T180), .(black, T181))
U17_gaggaa(T103, T113, T104, T107, T106, T108, unif_pairs51_out_ga(T103, T113)) → p50_out_gaggaa(T103, T113, T104, T107, T106, T108)
p50_in_gaggaa(T103, T113, T117, T118, T119, T120) → U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_in_ga(T103, T113))
U18_gaggaa(T103, T113, T117, T118, T119, T120, unif_pairs51_out_ga(T103, T113)) → U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_in_ga(.(T117, T118), .(T119, T120)))
unif_lines43_in_ga(.(T192, []), .(T193, [])) → unif_lines43_out_ga(.(T192, []), .(T193, []))
U19_gaggaa(T103, T113, T117, T118, T119, T120, unif_lines43_out_ga(.(T117, T118), .(T119, T120))) → p50_out_gaggaa(T103, T113, T117, T118, T119, T120)
U9_ga(T103, T104, T107, T105, T106, T108, p50_out_gaggaa(T103, .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, []))))))))))), T104, T107, T106, T108)) → unif_lines43_out_ga(.(T103, .(T104, T107)), .(T105, .(T106, T108)))
U6_ga(T56, T57, T58, unif_lines43_out_ga(T56, T57)) → unif_matrx37_out_ga(T56, .(T57, T58))
unif_matrx37_in_ga(T56, .(T65, T66)) → U7_ga(T56, T65, T66, unif_lines43_in_ga(T56, T65))
U7_ga(T56, T65, T66, unif_lines43_out_ga(T56, T65)) → U8_ga(T56, T65, T66, unif_matrx37_in_aa(T65, T66))
U8_ga(T56, T65, T66, unif_matrx37_out_aa(T65, T66)) → unif_matrx37_out_ga(T56, .(T65, T66))
unif_matrx37_in_ga(T199, []) → unif_matrx37_out_ga(T199, [])
U16_gag(T16, T37, T17, unif_matrx37_out_ga(T17, T37)) → p17_out_gag(T16, T37, T17)
U27_ga(T204, T6, p17_out_gag(T204, X293, black)) → shapes1_out_ga(.(black, T204), T6)
shapes1_in_ga(.(white, T207), T6) → U28_ga(T207, T6, p17_in_gag(T207, X310, w(X309)))
U28_ga(T207, T6, p17_out_gag(T207, X310, w(X309))) → shapes1_out_ga(.(white, T207), T6)

(99) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(100) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

VARMAT16_IN_GA(.(T26, T27), .(T28, X84)) → U2_GA(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
U2_GA(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → VARMAT16_IN_GA(T27, X84)
VARMAT16_IN_GA(.(T26, T27), .(X83, X84)) → VARMAT16_IN_GA(T26, X83)
VARMAT16_IN_GA(.(black, T33), .(black, X103)) → VARMAT16_IN_GA(T33, X103)
VARMAT16_IN_GA(.(white, T36), .(w(X119), X120)) → VARMAT16_IN_GA(T36, X120)

The TRS R consists of the following rules:

varmat16_in_ga([], []) → varmat16_out_ga([], [])
varmat16_in_ga(.(T26, T27), .(X83, X84)) → U1_ga(T26, T27, X83, X84, varmat16_in_ga(T26, X83))
varmat16_in_ga(.(T26, T27), .(T28, X84)) → U2_ga(T26, T27, T28, X84, varmat16_in_ga(T26, T28))
varmat16_in_ga(.(black, T33), .(black, X103)) → U4_ga(T33, X103, varmat16_in_ga(T33, X103))
varmat16_in_ga(.(white, T36), .(w(X119), X120)) → U5_ga(T36, X119, X120, varmat16_in_ga(T36, X120))
U1_ga(T26, T27, X83, X84, varmat16_out_ga(T26, X83)) → varmat16_out_ga(.(T26, T27), .(X83, X84))
U2_ga(T26, T27, T28, X84, varmat16_out_ga(T26, T28)) → U3_ga(T26, T27, T28, X84, varmat16_in_ga(T27, X84))
U4_ga(T33, X103, varmat16_out_ga(T33, X103)) → varmat16_out_ga(.(black, T33), .(black, X103))
U5_ga(T36, X119, X120, varmat16_out_ga(T36, X120)) → varmat16_out_ga(.(white, T36), .(w(X119), X120))
U3_ga(T26, T27, T28, X84, varmat16_out_ga(T27, X84)) → varmat16_out_ga(.(T26, T27), .(T28, X84))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
varmat16_in_ga(x1, x2) = varmat16_in_ga(x1)
[] = []
varmat16_out_ga(x1, x2) = varmat16_out_ga
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x2, x5)
black = black
U4_ga(x1, x2, x3) = U4_ga(x3)
white = white
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
w(x1) = w
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5)
VARMAT16_IN_GA(x1, x2) = VARMAT16_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x2, x5)

We have to consider all (P,R,Pi)-chains

(101) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(102) Obligation:

Q DP problem:
The TRS P consists of the following rules:

VARMAT16_IN_GA(.(T26, T27)) → U2_GA(T27, varmat16_in_ga(T26))
U2_GA(T27, varmat16_out_ga) → VARMAT16_IN_GA(T27)
VARMAT16_IN_GA(.(T26, T27)) → VARMAT16_IN_GA(T26)
VARMAT16_IN_GA(.(black, T33)) → VARMAT16_IN_GA(T33)
VARMAT16_IN_GA(.(white, T36)) → VARMAT16_IN_GA(T36)

The TRS R consists of the following rules:

varmat16_in_ga([]) → varmat16_out_ga
varmat16_in_ga(.(T26, T27)) → U1_ga(varmat16_in_ga(T26))
varmat16_in_ga(.(T26, T27)) → U2_ga(T27, varmat16_in_ga(T26))
varmat16_in_ga(.(black, T33)) → U4_ga(varmat16_in_ga(T33))
varmat16_in_ga(.(white, T36)) → U5_ga(varmat16_in_ga(T36))
U1_ga(varmat16_out_ga) → varmat16_out_ga
U2_ga(T27, varmat16_out_ga) → U3_ga(varmat16_in_ga(T27))
U4_ga(varmat16_out_ga) → varmat16_out_ga
U5_ga(varmat16_out_ga) → varmat16_out_ga
U3_ga(varmat16_out_ga) → varmat16_out_ga

The set Q consists of the following terms:

varmat16_in_ga(x0)
U1_ga(x0)
U2_ga(x0, x1)
U4_ga(x0)
U5_ga(x0)
U3_ga(x0)

We have to consider all (P,Q,R)-chains.

(103) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

U2_GA(T27, varmat16_out_ga) → VARMAT16_IN_GA(T27)
The graph contains the following edges 1 >= 1
VARMAT16_IN_GA(.(T26, T27)) → U2_GA(T27, varmat16_in_ga(T26))
The graph contains the following edges 1 > 1
VARMAT16_IN_GA(.(T26, T27)) → VARMAT16_IN_GA(T26)
The graph contains the following edges 1 > 1
VARMAT16_IN_GA(.(black, T33)) → VARMAT16_IN_GA(T33)
The graph contains the following edges 1 > 1
VARMAT16_IN_GA(.(white, T36)) → VARMAT16_IN_GA(T36)
The graph contains the following edges 1 > 1