Left Termination proof of /tmp/tmpFQDnW

Left Termination of the query pattern
transpose(g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 383 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 90 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇒, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 PiDP

        ↳15 PiDPToQDPProof (⇒, 0 ms)

        ↳16 QDP

        ↳17 QDPSizeChangeProof (⇔, 0 ms)

        ↳18 YES

(0) Obligation:

Clauses:

transpose(A, B) :- transpose_aux(A, [], B).
transpose_aux(.(R, Rs), X1, .(C, Cs)) :- ','(row2col(R, .(C, Cs), Cols1, [], Accm), transpose_aux(Rs, Accm, Cols1)).
transpose_aux([], X, X).
row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) :- row2col(Xs, Cols, Cols1, .([], A), B).
row2col([], [], [], A, A).

Query: transpose(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="D: transpose(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: transpose(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: transpose_aux(T5, [], T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 1 | transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="B: ','(row2col(T24, .(T28, T29), X35, [], X36), transpose_aux(T25, X36, X35))\n\nT24 is ground\nT25 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="C: row2col(T24, .(T28, T29), X35, [], X36)\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: transpose_aux(T25, T35, T34)\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: row2col(T24, .(T28, T29), X35, [], X36), SCOPE: 3, CLAUSE: 3 | row2col(T24, .(T28, T29), X35, [], X36), SCOPE: 3, CLAUSE: 4\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: row2col(T24, .(T28, T29), X35, [], X36), SCOPE: 3, CLAUSE: 3\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: row2col(T24, .(T28, T29), X35, [], X36), SCOPE: 3, CLAUSE: 4\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: row2col(T57, T60, X98, .([], []), X99)\n\nT57 is ground\nX98 is free\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: row2col(T57, T60, X98, .([], []), X99), SCOPE: 4, CLAUSE: 3 | row2col(T57, T60, X98, .([], []), X99), SCOPE: 4, CLAUSE: 4\n\nT57 is ground\nX98 is free\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: row2col(T57, T60, X98, .([], []), X99), SCOPE: 4, CLAUSE: 3\n\nT57 is ground\nX98 is free\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: row2col(T57, T60, X98, .([], []), X99), SCOPE: 4, CLAUSE: 4\n\nT57 is ground\nX98 is free\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: row2col(T82, T85, X152, .([], .([], [])), X153)\n\nT82 is ground\nX152 is free\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: row2col(T82, T85, X152, .([], .([], [])), X153), SCOPE: 5, CLAUSE: 3 | row2col(T82, T85, X152, .([], .([], [])), X153), SCOPE: 5, CLAUSE: 4\n\nT82 is ground\nX152 is free\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: row2col(T82, T85, X152, .([], .([], [])), X153), SCOPE: 5, CLAUSE: 3\n\nT82 is ground\nX152 is free\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: row2col(T82, T85, X152, .([], .([], [])), X153), SCOPE: 5, CLAUSE: 4\n\nT82 is ground\nX152 is free\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: row2col(T107, T110, X206, .([], .([], .([], []))), X207)\n\nT107 is ground\nX206 is free\nX207 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: row2col(T107, T110, X206, .([], .([], .([], []))), X207), SCOPE: 6, CLAUSE: 3 | row2col(T107, T110, X206, .([], .([], .([], []))), X207), SCOPE: 6, CLAUSE: 4\n\nT107 is ground\nX206 is free\nX207 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: row2col(T107, T110, X206, .([], .([], .([], []))), X207), SCOPE: 6, CLAUSE: 3\n\nT107 is ground\nX206 is free\nX207 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: row2col(T107, T110, X206, .([], .([], .([], []))), X207), SCOPE: 6, CLAUSE: 4\n\nT107 is ground\nX206 is free\nX207 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: row2col(T132, T135, X260, .([], .([], .([], .([], [])))), X261)\n\nT132 is ground\nX260 is free\nX261 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: row2col(T132, T135, X260, .([], .([], .([], .([], [])))), X261), SCOPE: 7, CLAUSE: 3 | row2col(T132, T135, X260, .([], .([], .([], .([], [])))), X261), SCOPE: 7, CLAUSE: 4\n\nT132 is ground\nX260 is free\nX261 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: row2col(T132, T135, X260, .([], .([], .([], .([], [])))), X261), SCOPE: 7, CLAUSE: 3\n\nT132 is ground\nX260 is free\nX261 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: row2col(T132, T135, X260, .([], .([], .([], .([], [])))), X261), SCOPE: 7, CLAUSE: 4\n\nT132 is ground\nX260 is free\nX261 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: row2col(T157, T160, X314, .([], .([], .([], .([], .([], []))))), X315)\n\nT157 is ground\nX314 is free\nX315 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: row2col(T157, T160, X314, .([], .([], .([], .([], .([], []))))), X315), SCOPE: 8, CLAUSE: 3 | row2col(T157, T160, X314, .([], .([], .([], .([], .([], []))))), X315), SCOPE: 8, CLAUSE: 4\n\nT157 is ground\nX314 is free\nX315 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: row2col(T157, T160, X314, .([], .([], .([], .([], .([], []))))), X315), SCOPE: 8, CLAUSE: 3\n\nT157 is ground\nX314 is free\nX315 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: row2col(T157, T160, X314, .([], .([], .([], .([], .([], []))))), X315), SCOPE: 8, CLAUSE: 4\n\nT157 is ground\nX314 is free\nX315 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: row2col(T182, T185, X368, .([], .([], .([], .([], .([], .([], [])))))), X369)\n\nT182 is ground\nX368 is free\nX369 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: row2col(T182, T185, X368, .([], .([], .([], .([], .([], .([], [])))))), X369), SCOPE: 9, CLAUSE: 3 | row2col(T182, T185, X368, .([], .([], .([], .([], .([], .([], [])))))), X369), SCOPE: 9, CLAUSE: 4\n\nT182 is ground\nX368 is free\nX369 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: row2col(T182, T185, X368, .([], .([], .([], .([], .([], .([], [])))))), X369), SCOPE: 9, CLAUSE: 3\n\nT182 is ground\nX368 is free\nX369 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: row2col(T182, T185, X368, .([], .([], .([], .([], .([], .([], [])))))), X369), SCOPE: 9, CLAUSE: 4\n\nT182 is ground\nX368 is free\nX369 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: row2col(T207, T210, X422, .([], .([], .([], .([], .([], .([], .([], []))))))), X423)\n\nT207 is ground\nX422 is free\nX423 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: row2col(T207, T210, X422, .([], .([], .([], .([], .([], .([], .([], []))))))), X423), SCOPE: 10, CLAUSE: 3 | row2col(T207, T210, X422, .([], .([], .([], .([], .([], .([], .([], []))))))), X423), SCOPE: 10, CLAUSE: 4\n\nT207 is ground\nX422 is free\nX423 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: row2col(T207, T210, X422, .([], .([], .([], .([], .([], .([], .([], []))))))), X423), SCOPE: 10, CLAUSE: 3\n\nT207 is ground\nX422 is free\nX423 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: row2col(T207, T210, X422, .([], .([], .([], .([], .([], .([], .([], []))))))), X423), SCOPE: 10, CLAUSE: 4\n\nT207 is ground\nX422 is free\nX423 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: row2col(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], .([], [])))))))), X477)\n\nT232 is ground\nX476 is free\nX477 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="A: row2col(T232, T235, X476, .([], T240), X477)\n\nT232 is ground\nT240 is ground\nX476 is free\nX477 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: row2col(T232, T235, X476, .([], T240), X477), SCOPE: 11, CLAUSE: 3 | row2col(T232, T235, X476, .([], T240), X477), SCOPE: 11, CLAUSE: 4\n\nT232 is ground\nT240 is ground\nX476 is free\nX477 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: row2col(T232, T235, X476, .([], T240), X477), SCOPE: 11, CLAUSE: 3\n\nT232 is ground\nT240 is ground\nX476 is free\nX477 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: row2col(T232, T235, X476, .([], T240), X477), SCOPE: 11, CLAUSE: 4\n\nT232 is ground\nT240 is ground\nX476 is free\nX477 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: row2col(T267, T271, X539, .([], .([], T270)), X540)\n\nT267 is ground\nT270 is ground\nX539 is free\nX540 is free", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: transpose_aux(T25, T35, T34), SCOPE: 12, CLAUSE: 1 | transpose_aux(T25, T35, T34), SCOPE: 12, CLAUSE: 2\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: transpose_aux(T25, T35, T34), SCOPE: 12, CLAUSE: 1\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: transpose_aux(T25, T35, T34), SCOPE: 12, CLAUSE: 2\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(row2col(T305, .(T310, T311), X611, [], X612), transpose_aux(T306, X612, X611))\n\nT305 is ground\nT306 is ground\nX611 is free\nX612 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 93 [arrowhead = none ];
93 -> 2;

94 [label="ONLY EVAL with clause\ntranspose(X4, X5) :- transpose_aux(X4, [], X5).\nand substitutionT1 -> T5,\nX4 -> T5,\nT2 -> T7,\nX5 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 94 [arrowhead = none ];
94 -> 3;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 95 [arrowhead = none ];
95 -> 4;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 96 [arrowhead = none ];
96 -> 5;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 97 [arrowhead = none ];
97 -> 6;

98 [label="EVAL with clause\ntranspose_aux(.(X30, X31), X32, .(X33, X34)) :- ','(row2col(X30, .(X33, X34), X35, [], X36), transpose_aux(X31, X36, X35)).\nand substitutionX30 -> T24,\nX31 -> T25,\nT5 -> .(T24, T25),\nX32 -> [],\nX33 -> T28,\nX34 -> T29,\nT7 -> .(T28, T29),\nT26 -> T28,\nT27 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 98 [arrowhead = none ];
98 -> 7;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 99 [arrowhead = none ];
99 -> 8;

100 [label="EVAL with clause\ntranspose_aux([], X627, X627).\nand substitutionT5 -> [],\nX627 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 100 [arrowhead = none ];
100 -> 90;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 101 [arrowhead = none ];
101 -> 91;

102 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 102 [arrowhead = none ];
102 -> 9;

103 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT35 is ground\nreplacements:X35 -> T34,\nX36 -> T35", fontsize=14, style = filled, fillcolor = violet];
7 -> 103 [arrowhead = none ];
103 -> 10;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 104 [arrowhead = none ];
104 -> 11;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 105 [arrowhead = none ];
105 -> 82;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 106 [arrowhead = none ];
106 -> 12;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 107 [arrowhead = none ];
107 -> 13;

108 [label="EVAL with clause\nrow2col(.(X91, X92), .(.(X91, X93), X94), .(X93, X95), X96, X97) :- row2col(X92, X94, X95, .([], X96), X97).\nand substitutionX91 -> T56,\nX92 -> T57,\nT24 -> .(T56, T57),\nX93 -> T58,\nT28 -> .(T56, T58),\nT29 -> T60,\nX94 -> T60,\nX95 -> X98,\nX35 -> .(T58, X98),\nX96 -> [],\nX36 -> X99,\nX97 -> X99,\nT59 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 108 [arrowhead = none ];
108 -> 14;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 109 [arrowhead = none ];
109 -> 15;

110 [label="BACKTRACK\nfor clause: row2col([], [], [], A, A)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 110 [arrowhead = none ];
110 -> 81;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 111 [arrowhead = none ];
111 -> 16;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 112 [arrowhead = none ];
112 -> 17;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 113 [arrowhead = none ];
113 -> 18;

114 [label="EVAL with clause\nrow2col(.(X145, X146), .(.(X145, X147), X148), .(X147, X149), X150, X151) :- row2col(X146, X148, X149, .([], X150), X151).\nand substitutionX145 -> T81,\nX146 -> T82,\nT57 -> .(T81, T82),\nX147 -> T83,\nX148 -> T85,\nT60 -> .(.(T81, T83), T85),\nX149 -> X152,\nX98 -> .(T83, X152),\nX150 -> .([], []),\nX99 -> X153,\nX151 -> X153,\nT84 -> T85", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 114 [arrowhead = none ];
114 -> 19;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 115 [arrowhead = none ];
115 -> 20;

116 [label="EVAL with clause\nrow2col([], [], [], X573, X573).\nand substitutionT57 -> [],\nT60 -> [],\nX98 -> [],\nX573 -> .([], []),\nX99 -> .([], [])", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 116 [arrowhead = none ];
116 -> 78;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 117 [arrowhead = none ];
117 -> 79;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 118 [arrowhead = none ];
118 -> 21;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 119 [arrowhead = none ];
119 -> 22;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 120 [arrowhead = none ];
120 -> 23;

121 [label="EVAL with clause\nrow2col(.(X199, X200), .(.(X199, X201), X202), .(X201, X203), X204, X205) :- row2col(X200, X202, X203, .([], X204), X205).\nand substitutionX199 -> T106,\nX200 -> T107,\nT82 -> .(T106, T107),\nX201 -> T108,\nX202 -> T110,\nT85 -> .(.(T106, T108), T110),\nX203 -> X206,\nX152 -> .(T108, X206),\nX204 -> .([], .([], [])),\nX153 -> X207,\nX205 -> X207,\nT109 -> T110", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 121 [arrowhead = none ];
121 -> 24;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 122 [arrowhead = none ];
122 -> 25;

123 [label="EVAL with clause\nrow2col([], [], [], X570, X570).\nand substitutionT82 -> [],\nT85 -> [],\nX152 -> [],\nX570 -> .([], .([], [])),\nX153 -> .([], .([], []))", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 123 [arrowhead = none ];
123 -> 75;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 124 [arrowhead = none ];
124 -> 76;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 125 [arrowhead = none ];
125 -> 26;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 126 [arrowhead = none ];
126 -> 27;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 127 [arrowhead = none ];
127 -> 28;

128 [label="EVAL with clause\nrow2col(.(X253, X254), .(.(X253, X255), X256), .(X255, X257), X258, X259) :- row2col(X254, X256, X257, .([], X258), X259).\nand substitutionX253 -> T131,\nX254 -> T132,\nT107 -> .(T131, T132),\nX255 -> T133,\nX256 -> T135,\nT110 -> .(.(T131, T133), T135),\nX257 -> X260,\nX206 -> .(T133, X260),\nX258 -> .([], .([], .([], []))),\nX207 -> X261,\nX259 -> X261,\nT134 -> T135", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 128 [arrowhead = none ];
128 -> 29;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 129 [arrowhead = none ];
129 -> 30;

130 [label="EVAL with clause\nrow2col([], [], [], X567, X567).\nand substitutionT107 -> [],\nT110 -> [],\nX206 -> [],\nX567 -> .([], .([], .([], []))),\nX207 -> .([], .([], .([], [])))", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 130 [arrowhead = none ];
130 -> 72;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 131 [arrowhead = none ];
131 -> 73;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 132 [arrowhead = none ];
132 -> 31;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 133 [arrowhead = none ];
133 -> 32;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 134 [arrowhead = none ];
134 -> 33;

135 [label="EVAL with clause\nrow2col(.(X307, X308), .(.(X307, X309), X310), .(X309, X311), X312, X313) :- row2col(X308, X310, X311, .([], X312), X313).\nand substitutionX307 -> T156,\nX308 -> T157,\nT132 -> .(T156, T157),\nX309 -> T158,\nX310 -> T160,\nT135 -> .(.(T156, T158), T160),\nX311 -> X314,\nX260 -> .(T158, X314),\nX312 -> .([], .([], .([], .([], [])))),\nX261 -> X315,\nX313 -> X315,\nT159 -> T160", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 135 [arrowhead = none ];
135 -> 34;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 136 [arrowhead = none ];
136 -> 35;

137 [label="EVAL with clause\nrow2col([], [], [], X564, X564).\nand substitutionT132 -> [],\nT135 -> [],\nX260 -> [],\nX564 -> .([], .([], .([], .([], [])))),\nX261 -> .([], .([], .([], .([], []))))", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 137 [arrowhead = none ];
137 -> 69;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 138 [arrowhead = none ];
138 -> 70;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 139 [arrowhead = none ];
139 -> 36;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 140 [arrowhead = none ];
140 -> 37;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 141 [arrowhead = none ];
141 -> 38;

142 [label="EVAL with clause\nrow2col(.(X361, X362), .(.(X361, X363), X364), .(X363, X365), X366, X367) :- row2col(X362, X364, X365, .([], X366), X367).\nand substitutionX361 -> T181,\nX362 -> T182,\nT157 -> .(T181, T182),\nX363 -> T183,\nX364 -> T185,\nT160 -> .(.(T181, T183), T185),\nX365 -> X368,\nX314 -> .(T183, X368),\nX366 -> .([], .([], .([], .([], .([], []))))),\nX315 -> X369,\nX367 -> X369,\nT184 -> T185", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 142 [arrowhead = none ];
142 -> 39;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 143 [arrowhead = none ];
143 -> 40;

144 [label="EVAL with clause\nrow2col([], [], [], X561, X561).\nand substitutionT157 -> [],\nT160 -> [],\nX314 -> [],\nX561 -> .([], .([], .([], .([], .([], []))))),\nX315 -> .([], .([], .([], .([], .([], [])))))", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 144 [arrowhead = none ];
144 -> 66;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 145 [arrowhead = none ];
145 -> 67;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 146 [arrowhead = none ];
146 -> 41;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 147 [arrowhead = none ];
147 -> 42;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 148 [arrowhead = none ];
148 -> 43;

149 [label="EVAL with clause\nrow2col(.(X415, X416), .(.(X415, X417), X418), .(X417, X419), X420, X421) :- row2col(X416, X418, X419, .([], X420), X421).\nand substitutionX415 -> T206,\nX416 -> T207,\nT182 -> .(T206, T207),\nX417 -> T208,\nX418 -> T210,\nT185 -> .(.(T206, T208), T210),\nX419 -> X422,\nX368 -> .(T208, X422),\nX420 -> .([], .([], .([], .([], .([], .([], [])))))),\nX369 -> X423,\nX421 -> X423,\nT209 -> T210", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 149 [arrowhead = none ];
149 -> 44;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 150 [arrowhead = none ];
150 -> 45;

151 [label="EVAL with clause\nrow2col([], [], [], X558, X558).\nand substitutionT182 -> [],\nT185 -> [],\nX368 -> [],\nX558 -> .([], .([], .([], .([], .([], .([], [])))))),\nX369 -> .([], .([], .([], .([], .([], .([], []))))))", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 151 [arrowhead = none ];
151 -> 63;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 152 [arrowhead = none ];
152 -> 64;

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 153 [arrowhead = none ];
153 -> 46;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 154 [arrowhead = none ];
154 -> 47;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 155 [arrowhead = none ];
155 -> 48;

156 [label="EVAL with clause\nrow2col(.(X469, X470), .(.(X469, X471), X472), .(X471, X473), X474, X475) :- row2col(X470, X472, X473, .([], X474), X475).\nand substitutionX469 -> T231,\nX470 -> T232,\nT207 -> .(T231, T232),\nX471 -> T233,\nX472 -> T235,\nT210 -> .(.(T231, T233), T235),\nX473 -> X476,\nX422 -> .(T233, X476),\nX474 -> .([], .([], .([], .([], .([], .([], .([], []))))))),\nX423 -> X477,\nX475 -> X477,\nT234 -> T235", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 156 [arrowhead = none ];
156 -> 49;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 157 [arrowhead = none ];
157 -> 50;

158 [label="EVAL with clause\nrow2col([], [], [], X555, X555).\nand substitutionT207 -> [],\nT210 -> [],\nX422 -> [],\nX555 -> .([], .([], .([], .([], .([], .([], .([], []))))))),\nX423 -> .([], .([], .([], .([], .([], .([], .([], [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 158 [arrowhead = none ];
158 -> 60;

159 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 159 [arrowhead = none ];
159 -> 61;

160 [label="GENERALIZATION\nT240 <-- .([], .([], .([], .([], .([], .([], .([], [])))))))\n\nNew Knowledge:\nT240 is ground", fontsize=14, style = filled, fillcolor = violet];
49 -> 160 [arrowhead = none ];
160 -> 51;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 161 [arrowhead = none ];
161 -> 52;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 162 [arrowhead = none ];
162 -> 53;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 163 [arrowhead = none ];
163 -> 54;

164 [label="EVAL with clause\nrow2col(.(X532, X533), .(.(X532, X534), X535), .(X534, X536), X537, X538) :- row2col(X533, X535, X536, .([], X537), X538).\nand substitutionX532 -> T266,\nX533 -> T267,\nT232 -> .(T266, T267),\nX534 -> T268,\nX535 -> T271,\nT235 -> .(.(T266, T268), T271),\nX536 -> X539,\nX476 -> .(T268, X539),\nT240 -> T270,\nX537 -> .([], T270),\nX477 -> X540,\nX538 -> X540,\nT269 -> T271", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 164 [arrowhead = none ];
164 -> 55;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 165 [arrowhead = none ];
165 -> 56;

166 [label="EVAL with clause\nrow2col([], [], [], X552, X552).\nand substitutionT232 -> [],\nT235 -> [],\nX476 -> [],\nT240 -> T279,\nX552 -> .([], T279),\nX477 -> .([], T279)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 166 [arrowhead = none ];
166 -> 57;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 167 [arrowhead = none ];
167 -> 58;

168 [label="INSTANCE with matching:\nT232 -> T267\nT235 -> T271\nX476 -> X539\nT240 -> .([], T270)\nX477 -> X540", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 168 [arrowhead = none , style=dashed];
168 -> 51[style=dashed];

169 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 169 [arrowhead = none ];
169 -> 59;

170 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 170 [arrowhead = none ];
170 -> 62;

171 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 171 [arrowhead = none ];
171 -> 65;

172 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 172 [arrowhead = none ];
172 -> 68;

173 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 173 [arrowhead = none ];
173 -> 71;

174 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 174 [arrowhead = none ];
174 -> 74;

175 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 175 [arrowhead = none ];
175 -> 77;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 176 [arrowhead = none ];
176 -> 80;

177 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 177 [arrowhead = none ];
177 -> 83;

178 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 178 [arrowhead = none ];
178 -> 84;

179 [label="EVAL with clause\ntranspose_aux(.(X606, X607), X608, .(X609, X610)) :- ','(row2col(X606, .(X609, X610), X611, [], X612), transpose_aux(X607, X612, X611)).\nand substitutionX606 -> T305,\nX607 -> T306,\nT25 -> .(T305, T306),\nT35 -> T307,\nX608 -> T307,\nX609 -> T310,\nX610 -> T311,\nT34 -> .(T310, T311),\nT308 -> T310,\nT309 -> T311", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 179 [arrowhead = none ];
179 -> 85;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 180 [arrowhead = none ];
180 -> 86;

181 [label="EVAL with clause\ntranspose_aux([], X624, X624).\nand substitutionT25 -> [],\nT35 -> T318,\nX624 -> T318,\nT34 -> T318", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 181 [arrowhead = none ];
181 -> 87;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 182 [arrowhead = none ];
182 -> 88;

183 [label="INSTANCE with matching:\nT24 -> T305\nT28 -> T310\nT29 -> T311\nX35 -> X611\nX36 -> X612\nT25 -> T306", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 183 [arrowhead = none , style=dashed];
183 -> 7[style=dashed];

184 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 184 [arrowhead = none ];
184 -> 89;

185 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 185 [arrowhead = none ];
185 -> 92;

}

(2) Obligation:

Triples:

row2colA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) :- row2colA(X2, X4, X5, .([], X6), X7).
pB(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20, X21) :- row2colA(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20).
pB(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) :- ','(row2colcC(X1, X2, X3, .(X4, X5), X6), pB(X7, X4, X5, X9, X10, X8)).
transposeD(.(X1, X2), .(X3, X4)) :- pB(X1, X3, X4, X5, X6, X2).

Clauses:

row2colcA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) :- row2colcA(X2, X4, X5, .([], X6), X7).
row2colcA([], [], [], X1, .([], X1)).
qcB(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) :- ','(row2colcC(X1, X2, X3, .(X4, X5), X6), qcB(X7, X4, X5, X9, X10, X8)).
qcB(X1, X2, X3, X4, X4, []) :- row2colcC(X1, X2, X3, X4, X4).
row2colcC(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20) :- row2colcA(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20).
row2colcC(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X1, X8), .(.(X2, X9), .(.(X3, X10), .(.(X4, X11), .(.(X5, X12), .(.(X6, X13), .(.(X7, X14), [])))))), .(X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))).
row2colcC(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X1, X7), .(.(X2, X8), .(.(X3, X9), .(.(X4, X10), .(.(X5, X11), .(.(X6, X12), []))))), .(X7, .(X8, .(X9, .(X10, .(X11, .(X12, [])))))), .([], .([], .([], .([], .([], .([], []))))))).
row2colcC(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X1, X6), .(.(X2, X7), .(.(X3, X8), .(.(X4, X9), .(.(X5, X10), [])))), .(X6, .(X7, .(X8, .(X9, .(X10, []))))), .([], .([], .([], .([], .([], [])))))).
row2colcC(.(X1, .(X2, .(X3, .(X4, [])))), .(X1, X5), .(.(X2, X6), .(.(X3, X7), .(.(X4, X8), []))), .(X5, .(X6, .(X7, .(X8, [])))), .([], .([], .([], .([], []))))).
row2colcC(.(X1, .(X2, .(X3, []))), .(X1, X4), .(.(X2, X5), .(.(X3, X6), [])), .(X4, .(X5, .(X6, []))), .([], .([], .([], [])))).
row2colcC(.(X1, .(X2, [])), .(X1, X3), .(.(X2, X4), []), .(X3, .(X4, [])), .([], .([], []))).
row2colcC(.(X1, []), .(X1, X2), [], .(X2, []), .([], [])).

Afs:

transposeD(x1, x2) = transposeD(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
transposeD_in: (b,f)
pB_in: (b,f,f,f,f,b)
row2colA_in: (b,f,f,b,f)
row2colcC_in: (b,f,f,f,f)
row2colcA_in: (b,f,f,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSED_IN_GA(.(X1, X2), .(X3, X4)) → U5_GA(X1, X2, X3, X4, pB_in_gaaaag(X1, X3, X4, X5, X6, X2))
TRANSPOSED_IN_GA(.(X1, X2), .(X3, X4)) → PB_IN_GAAAAG(X1, X3, X4, X5, X6, X2)
PB_IN_GAAAAG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20, X21) → U2_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, row2colA_in_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20))
PB_IN_GAAAAG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20, X21) → ROW2COLA_IN_GAAGA(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20)
ROW2COLA_IN_GAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → U1_GAAGA(X1, X2, X3, X4, X5, X6, X7, row2colA_in_gaaga(X2, X4, X5, .([], X6), X7))
ROW2COLA_IN_GAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → ROW2COLA_IN_GAAGA(X2, X4, X5, .([], X6), X7)
PB_IN_GAAAAG(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) → U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_gaaaa(X1, X2, X3, .(X4, X5), X6))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → U4_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, pB_in_gaaaag(X7, X4, X5, X9, X10, X8))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_GAAAAG(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20) → U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colcA_in_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20))
row2colcA_in_gaaga(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → U7_gaaga(X1, X2, X3, X4, X5, X6, X7, row2colcA_in_gaaga(X2, X4, X5, .([], X6), X7))
row2colcA_in_gaaga([], [], [], X1, .([], X1)) → row2colcA_out_gaaga([], [], [], X1, .([], X1))
U7_gaaga(X1, X2, X3, X4, X5, X6, X7, row2colcA_out_gaaga(X2, X4, X5, .([], X6), X7)) → row2colcA_out_gaaga(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7)
U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colcA_out_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20)) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20)
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X1, X8), .(.(X2, X9), .(.(X3, X10), .(.(X4, X11), .(.(X5, X12), .(.(X6, X13), .(.(X7, X14), [])))))), .(X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X1, X8), .(.(X2, X9), .(.(X3, X10), .(.(X4, X11), .(.(X5, X12), .(.(X6, X13), .(.(X7, X14), [])))))), .(X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X1, X7), .(.(X2, X8), .(.(X3, X9), .(.(X4, X10), .(.(X5, X11), .(.(X6, X12), []))))), .(X7, .(X8, .(X9, .(X10, .(X11, .(X12, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X1, X7), .(.(X2, X8), .(.(X3, X9), .(.(X4, X10), .(.(X5, X11), .(.(X6, X12), []))))), .(X7, .(X8, .(X9, .(X10, .(X11, .(X12, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X1, X6), .(.(X2, X7), .(.(X3, X8), .(.(X4, X9), .(.(X5, X10), [])))), .(X6, .(X7, .(X8, .(X9, .(X10, []))))), .([], .([], .([], .([], .([], [])))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X1, X6), .(.(X2, X7), .(.(X3, X8), .(.(X4, X9), .(.(X5, X10), [])))), .(X6, .(X7, .(X8, .(X9, .(X10, []))))), .([], .([], .([], .([], .([], []))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, [])))), .(X1, X5), .(.(X2, X6), .(.(X3, X7), .(.(X4, X8), []))), .(X5, .(X6, .(X7, .(X8, [])))), .([], .([], .([], .([], []))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, [])))), .(X1, X5), .(.(X2, X6), .(.(X3, X7), .(.(X4, X8), []))), .(X5, .(X6, .(X7, .(X8, [])))), .([], .([], .([], .([], [])))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, []))), .(X1, X4), .(.(X2, X5), .(.(X3, X6), [])), .(X4, .(X5, .(X6, []))), .([], .([], .([], [])))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, []))), .(X1, X4), .(.(X2, X5), .(.(X3, X6), [])), .(X4, .(X5, .(X6, []))), .([], .([], .([], []))))
row2colcC_in_gaaaa(.(X1, .(X2, [])), .(X1, X3), .(.(X2, X4), []), .(X3, .(X4, [])), .([], .([], []))) → row2colcC_out_gaaaa(.(X1, .(X2, [])), .(X1, X3), .(.(X2, X4), []), .(X3, .(X4, [])), .([], .([], [])))
row2colcC_in_gaaaa(.(X1, []), .(X1, X2), [], .(X2, []), .([], [])) → row2colcC_out_gaaaa(.(X1, []), .(X1, X2), [], .(X2, []), .([], []))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6) = pB_in_gaaaag(x1, x6)
row2colA_in_gaaga(x1, x2, x3, x4, x5) = row2colA_in_gaaga(x1, x4)
[] = []
row2colcC_in_gaaaa(x1, x2, x3, x4, x5) = row2colcC_in_gaaaa(x1)
U11_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U11_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x21)
row2colcA_in_gaaga(x1, x2, x3, x4, x5) = row2colcA_in_gaaga(x1, x4)
U7_gaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaaga(x1, x2, x6, x8)
row2colcA_out_gaaga(x1, x2, x3, x4, x5) = row2colcA_out_gaaga(x1, x4, x5)
row2colcC_out_gaaaa(x1, x2, x3, x4, x5) = row2colcC_out_gaaaa(x1, x5)
TRANSPOSED_IN_GA(x1, x2) = TRANSPOSED_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5) = U5_GA(x1, x2, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6) = PB_IN_GAAAAG(x1, x6)
U2_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22) = U2_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x21, x22)
ROW2COLA_IN_GAAGA(x1, x2, x3, x4, x5) = ROW2COLA_IN_GAAGA(x1, x4)
U1_GAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_GAAGA(x1, x2, x6, x8)
U3_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_GAAAAG(x1, x7, x8, x9)
U4_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_GAAAAG(x1, x6, x7, x8, x9)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

TRANSPOSED_IN_GA(.(X1, X2), .(X3, X4)) → U5_GA(X1, X2, X3, X4, pB_in_gaaaag(X1, X3, X4, X5, X6, X2))
TRANSPOSED_IN_GA(.(X1, X2), .(X3, X4)) → PB_IN_GAAAAG(X1, X3, X4, X5, X6, X2)
PB_IN_GAAAAG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20, X21) → U2_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, row2colA_in_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20))
PB_IN_GAAAAG(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20, X21) → ROW2COLA_IN_GAAGA(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20)
ROW2COLA_IN_GAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → U1_GAAGA(X1, X2, X3, X4, X5, X6, X7, row2colA_in_gaaga(X2, X4, X5, .([], X6), X7))
ROW2COLA_IN_GAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → ROW2COLA_IN_GAAGA(X2, X4, X5, .([], X6), X7)
PB_IN_GAAAAG(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) → U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_gaaaa(X1, X2, X3, .(X4, X5), X6))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → U4_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, pB_in_gaaaag(X7, X4, X5, X9, X10, X8))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_GAAAAG(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20) → U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colcA_in_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20))
row2colcA_in_gaaga(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → U7_gaaga(X1, X2, X3, X4, X5, X6, X7, row2colcA_in_gaaga(X2, X4, X5, .([], X6), X7))
row2colcA_in_gaaga([], [], [], X1, .([], X1)) → row2colcA_out_gaaga([], [], [], X1, .([], X1))
U7_gaaga(X1, X2, X3, X4, X5, X6, X7, row2colcA_out_gaaga(X2, X4, X5, .([], X6), X7)) → row2colcA_out_gaaga(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7)
U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colcA_out_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20)) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20)
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X1, X8), .(.(X2, X9), .(.(X3, X10), .(.(X4, X11), .(.(X5, X12), .(.(X6, X13), .(.(X7, X14), [])))))), .(X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X1, X8), .(.(X2, X9), .(.(X3, X10), .(.(X4, X11), .(.(X5, X12), .(.(X6, X13), .(.(X7, X14), [])))))), .(X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X1, X7), .(.(X2, X8), .(.(X3, X9), .(.(X4, X10), .(.(X5, X11), .(.(X6, X12), []))))), .(X7, .(X8, .(X9, .(X10, .(X11, .(X12, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X1, X7), .(.(X2, X8), .(.(X3, X9), .(.(X4, X10), .(.(X5, X11), .(.(X6, X12), []))))), .(X7, .(X8, .(X9, .(X10, .(X11, .(X12, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X1, X6), .(.(X2, X7), .(.(X3, X8), .(.(X4, X9), .(.(X5, X10), [])))), .(X6, .(X7, .(X8, .(X9, .(X10, []))))), .([], .([], .([], .([], .([], [])))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X1, X6), .(.(X2, X7), .(.(X3, X8), .(.(X4, X9), .(.(X5, X10), [])))), .(X6, .(X7, .(X8, .(X9, .(X10, []))))), .([], .([], .([], .([], .([], []))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, [])))), .(X1, X5), .(.(X2, X6), .(.(X3, X7), .(.(X4, X8), []))), .(X5, .(X6, .(X7, .(X8, [])))), .([], .([], .([], .([], []))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, [])))), .(X1, X5), .(.(X2, X6), .(.(X3, X7), .(.(X4, X8), []))), .(X5, .(X6, .(X7, .(X8, [])))), .([], .([], .([], .([], [])))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, []))), .(X1, X4), .(.(X2, X5), .(.(X3, X6), [])), .(X4, .(X5, .(X6, []))), .([], .([], .([], [])))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, []))), .(X1, X4), .(.(X2, X5), .(.(X3, X6), [])), .(X4, .(X5, .(X6, []))), .([], .([], .([], []))))
row2colcC_in_gaaaa(.(X1, .(X2, [])), .(X1, X3), .(.(X2, X4), []), .(X3, .(X4, [])), .([], .([], []))) → row2colcC_out_gaaaa(.(X1, .(X2, [])), .(X1, X3), .(.(X2, X4), []), .(X3, .(X4, [])), .([], .([], [])))
row2colcC_in_gaaaa(.(X1, []), .(X1, X2), [], .(X2, []), .([], [])) → row2colcC_out_gaaaa(.(X1, []), .(X1, X2), [], .(X2, []), .([], []))

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ROW2COLA_IN_GAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → ROW2COLA_IN_GAAGA(X2, X4, X5, .([], X6), X7)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20) → U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colcA_in_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20))
row2colcA_in_gaaga(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → U7_gaaga(X1, X2, X3, X4, X5, X6, X7, row2colcA_in_gaaga(X2, X4, X5, .([], X6), X7))
row2colcA_in_gaaga([], [], [], X1, .([], X1)) → row2colcA_out_gaaga([], [], [], X1, .([], X1))
U7_gaaga(X1, X2, X3, X4, X5, X6, X7, row2colcA_out_gaaga(X2, X4, X5, .([], X6), X7)) → row2colcA_out_gaaga(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7)
U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colcA_out_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20)) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20)
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X1, X8), .(.(X2, X9), .(.(X3, X10), .(.(X4, X11), .(.(X5, X12), .(.(X6, X13), .(.(X7, X14), [])))))), .(X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X1, X8), .(.(X2, X9), .(.(X3, X10), .(.(X4, X11), .(.(X5, X12), .(.(X6, X13), .(.(X7, X14), [])))))), .(X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X1, X7), .(.(X2, X8), .(.(X3, X9), .(.(X4, X10), .(.(X5, X11), .(.(X6, X12), []))))), .(X7, .(X8, .(X9, .(X10, .(X11, .(X12, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X1, X7), .(.(X2, X8), .(.(X3, X9), .(.(X4, X10), .(.(X5, X11), .(.(X6, X12), []))))), .(X7, .(X8, .(X9, .(X10, .(X11, .(X12, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X1, X6), .(.(X2, X7), .(.(X3, X8), .(.(X4, X9), .(.(X5, X10), [])))), .(X6, .(X7, .(X8, .(X9, .(X10, []))))), .([], .([], .([], .([], .([], [])))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X1, X6), .(.(X2, X7), .(.(X3, X8), .(.(X4, X9), .(.(X5, X10), [])))), .(X6, .(X7, .(X8, .(X9, .(X10, []))))), .([], .([], .([], .([], .([], []))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, [])))), .(X1, X5), .(.(X2, X6), .(.(X3, X7), .(.(X4, X8), []))), .(X5, .(X6, .(X7, .(X8, [])))), .([], .([], .([], .([], []))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, [])))), .(X1, X5), .(.(X2, X6), .(.(X3, X7), .(.(X4, X8), []))), .(X5, .(X6, .(X7, .(X8, [])))), .([], .([], .([], .([], [])))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, []))), .(X1, X4), .(.(X2, X5), .(.(X3, X6), [])), .(X4, .(X5, .(X6, []))), .([], .([], .([], [])))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, []))), .(X1, X4), .(.(X2, X5), .(.(X3, X6), [])), .(X4, .(X5, .(X6, []))), .([], .([], .([], []))))
row2colcC_in_gaaaa(.(X1, .(X2, [])), .(X1, X3), .(.(X2, X4), []), .(X3, .(X4, [])), .([], .([], []))) → row2colcC_out_gaaaa(.(X1, .(X2, [])), .(X1, X3), .(.(X2, X4), []), .(X3, .(X4, [])), .([], .([], [])))
row2colcC_in_gaaaa(.(X1, []), .(X1, X2), [], .(X2, []), .([], [])) → row2colcC_out_gaaaa(.(X1, []), .(X1, X2), [], .(X2, []), .([], []))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
row2colcC_in_gaaaa(x1, x2, x3, x4, x5) = row2colcC_in_gaaaa(x1)
U11_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U11_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x21)
row2colcA_in_gaaga(x1, x2, x3, x4, x5) = row2colcA_in_gaaga(x1, x4)
U7_gaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaaga(x1, x2, x6, x8)
row2colcA_out_gaaga(x1, x2, x3, x4, x5) = row2colcA_out_gaaga(x1, x4, x5)
row2colcC_out_gaaaa(x1, x2, x3, x4, x5) = row2colcC_out_gaaaa(x1, x5)
ROW2COLA_IN_GAAGA(x1, x2, x3, x4, x5) = ROW2COLA_IN_GAAGA(x1, x4)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

ROW2COLA_IN_GAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → ROW2COLA_IN_GAAGA(X2, X4, X5, .([], X6), X7)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COLA_IN_GAAGA(x1, x2, x3, x4, x5) = ROW2COLA_IN_GAAGA(x1, x4)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

ROW2COLA_IN_GAAGA(.(X1, X2), X6) → ROW2COLA_IN_GAAGA(X2, .([], X6))

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

(12) 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:

ROW2COLA_IN_GAAGA(.(X1, X2), X6) → ROW2COLA_IN_GAAGA(X2, .([], X6))
The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

PB_IN_GAAAAG(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) → U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_gaaaa(X1, X2, X3, .(X4, X5), X6))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_GAAAAG(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20) → U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colcA_in_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20))
row2colcA_in_gaaga(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) → U7_gaaga(X1, X2, X3, X4, X5, X6, X7, row2colcA_in_gaaga(X2, X4, X5, .([], X6), X7))
row2colcA_in_gaaga([], [], [], X1, .([], X1)) → row2colcA_out_gaaga([], [], [], X1, .([], X1))
U7_gaaga(X1, X2, X3, X4, X5, X6, X7, row2colcA_out_gaaga(X2, X4, X5, .([], X6), X7)) → row2colcA_out_gaaga(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7)
U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colcA_out_gaaga(X9, X18, X19, .([], .([], .([], .([], .([], .([], .([], []))))))), X20)) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), .(X1, X10), .(.(X2, X11), .(.(X3, X12), .(.(X4, X13), .(.(X5, X14), .(.(X6, X15), .(.(X7, X16), .(.(X8, X17), X18))))))), .(X10, .(X11, .(X12, .(X13, .(X14, .(X15, .(X16, .(X17, X19)))))))), X20)
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X1, X8), .(.(X2, X9), .(.(X3, X10), .(.(X4, X11), .(.(X5, X12), .(.(X6, X13), .(.(X7, X14), [])))))), .(X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .(X1, X8), .(.(X2, X9), .(.(X3, X10), .(.(X4, X11), .(.(X5, X12), .(.(X6, X13), .(.(X7, X14), [])))))), .(X8, .(X9, .(X10, .(X11, .(X12, .(X13, .(X14, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X1, X7), .(.(X2, X8), .(.(X3, X9), .(.(X4, X10), .(.(X5, X11), .(.(X6, X12), []))))), .(X7, .(X8, .(X9, .(X10, .(X11, .(X12, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .(X1, X7), .(.(X2, X8), .(.(X3, X9), .(.(X4, X10), .(.(X5, X11), .(.(X6, X12), []))))), .(X7, .(X8, .(X9, .(X10, .(X11, .(X12, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X1, X6), .(.(X2, X7), .(.(X3, X8), .(.(X4, X9), .(.(X5, X10), [])))), .(X6, .(X7, .(X8, .(X9, .(X10, []))))), .([], .([], .([], .([], .([], [])))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .(X1, X6), .(.(X2, X7), .(.(X3, X8), .(.(X4, X9), .(.(X5, X10), [])))), .(X6, .(X7, .(X8, .(X9, .(X10, []))))), .([], .([], .([], .([], .([], []))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, [])))), .(X1, X5), .(.(X2, X6), .(.(X3, X7), .(.(X4, X8), []))), .(X5, .(X6, .(X7, .(X8, [])))), .([], .([], .([], .([], []))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, [])))), .(X1, X5), .(.(X2, X6), .(.(X3, X7), .(.(X4, X8), []))), .(X5, .(X6, .(X7, .(X8, [])))), .([], .([], .([], .([], [])))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, []))), .(X1, X4), .(.(X2, X5), .(.(X3, X6), [])), .(X4, .(X5, .(X6, []))), .([], .([], .([], [])))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, []))), .(X1, X4), .(.(X2, X5), .(.(X3, X6), [])), .(X4, .(X5, .(X6, []))), .([], .([], .([], []))))
row2colcC_in_gaaaa(.(X1, .(X2, [])), .(X1, X3), .(.(X2, X4), []), .(X3, .(X4, [])), .([], .([], []))) → row2colcC_out_gaaaa(.(X1, .(X2, [])), .(X1, X3), .(.(X2, X4), []), .(X3, .(X4, [])), .([], .([], [])))
row2colcC_in_gaaaa(.(X1, []), .(X1, X2), [], .(X2, []), .([], [])) → row2colcC_out_gaaaa(.(X1, []), .(X1, X2), [], .(X2, []), .([], []))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
row2colcC_in_gaaaa(x1, x2, x3, x4, x5) = row2colcC_in_gaaaa(x1)
U11_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U11_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x21)
row2colcA_in_gaaga(x1, x2, x3, x4, x5) = row2colcA_in_gaaga(x1, x4)
U7_gaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gaaga(x1, x2, x6, x8)
row2colcA_out_gaaga(x1, x2, x3, x4, x5) = row2colcA_out_gaaga(x1, x4, x5)
row2colcC_out_gaaaa(x1, x2, x3, x4, x5) = row2colcC_out_gaaaa(x1, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6) = PB_IN_GAAAAG(x1, x6)
U3_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_GAAAAG(x1, x7, x8, x9)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

PB_IN_GAAAAG(X1, .(X7, X8)) → U3_GAAAAG(X1, X7, X8, row2colcC_in_gaaaa(X1))
U3_GAAAAG(X1, X7, X8, row2colcC_out_gaaaa(X1, X6)) → PB_IN_GAAAAG(X7, X8)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9))))))))) → U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, row2colcA_in_gaaga(X9, .([], .([], .([], .([], .([], .([], .([], [])))))))))
row2colcA_in_gaaga(.(X1, X2), X6) → U7_gaaga(X1, X2, X6, row2colcA_in_gaaga(X2, .([], X6)))
row2colcA_in_gaaga([], X1) → row2colcA_out_gaaga([], X1, .([], X1))
U7_gaaga(X1, X2, X6, row2colcA_out_gaaga(X2, .([], X6), X7)) → row2colcA_out_gaaga(.(X1, X2), X6, X7)
U11_gaaaa(X1, X2, X3, X4, X5, X6, X7, X8, X9, row2colcA_out_gaaga(X9, .([], .([], .([], .([], .([], .([], .([], []))))))), X20)) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X20)
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, [])))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, []))))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, [])))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, .(X5, []))))), .([], .([], .([], .([], .([], []))))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, .(X4, []))))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, .(X4, [])))), .([], .([], .([], .([], [])))))
row2colcC_in_gaaaa(.(X1, .(X2, .(X3, [])))) → row2colcC_out_gaaaa(.(X1, .(X2, .(X3, []))), .([], .([], .([], []))))
row2colcC_in_gaaaa(.(X1, .(X2, []))) → row2colcC_out_gaaaa(.(X1, .(X2, [])), .([], .([], [])))
row2colcC_in_gaaaa(.(X1, [])) → row2colcC_out_gaaaa(.(X1, []), .([], []))

The set Q consists of the following terms:

row2colcC_in_gaaaa(x0)
row2colcA_in_gaaga(x0, x1)
U7_gaaga(x0, x1, x2, x3)
U11_gaaaa(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)

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

(17) 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:

U3_GAAAAG(X1, X7, X8, row2colcC_out_gaaaa(X1, X6)) → PB_IN_GAAAAG(X7, X8)
The graph contains the following edges 2 >= 1, 3 >= 2
PB_IN_GAAAAG(X1, .(X7, X8)) → U3_GAAAAG(X1, X7, X8, row2colcC_in_gaaaa(X1))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) PiDPToQDPProof (SOUND transformation)

(16) Obligation:

(17) QDPSizeChangeProof (EQUIVALENT transformation)

(18) YES