Left Termination proof of /tmp/tmpD0mFew/transpose.pl

Left Termination of the query pattern transpose_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

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 QDPSizeChangeProof (⇔)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇐)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 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).

Queries:

transpose(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: 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="7: ','(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="9: 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, X103, .([], []), X104)\n\nT57 is ground\nX103 is free; \nX104 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, X103, .([], []), X104), SCOPE: 4, CLAUSE: 3 | row2col(T57, T60, X103, .([], []), X104), SCOPE: 4, CLAUSE: 4\n\nT57 is ground\nX103 is free; \nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: row2col(T57, T60, X103, .([], []), X104), SCOPE: 4, CLAUSE: 3\n\nT57 is ground\nX103 is free; \nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: row2col(T57, T60, X103, .([], []), X104), SCOPE: 4, CLAUSE: 4\n\nT57 is ground\nX103 is free; \nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: row2col(T82, T85, X164, .([], .([], [])), X165)\n\nT82 is ground\nX164 is free; \nX165 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, X164, .([], .([], [])), X165), SCOPE: 5, CLAUSE: 3 | row2col(T82, T85, X164, .([], .([], [])), X165), SCOPE: 5, CLAUSE: 4\n\nT82 is ground\nX164 is free; \nX165 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: row2col(T82, T85, X164, .([], .([], [])), X165), SCOPE: 5, CLAUSE: 3\n\nT82 is ground\nX164 is free; \nX165 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: row2col(T82, T85, X164, .([], .([], [])), X165), SCOPE: 5, CLAUSE: 4\n\nT82 is ground\nX164 is free; \nX165 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: row2col(T107, T110, X225, .([], .([], .([], []))), X226)\n\nT107 is ground\nX225 is free; \nX226 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, X225, .([], .([], .([], []))), X226), SCOPE: 6, CLAUSE: 3 | row2col(T107, T110, X225, .([], .([], .([], []))), X226), SCOPE: 6, CLAUSE: 4\n\nT107 is ground\nX225 is free; \nX226 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: row2col(T107, T110, X225, .([], .([], .([], []))), X226), SCOPE: 6, CLAUSE: 3\n\nT107 is ground\nX225 is free; \nX226 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: row2col(T107, T110, X225, .([], .([], .([], []))), X226), SCOPE: 6, CLAUSE: 4\n\nT107 is ground\nX225 is free; \nX226 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: row2col(T132, T135, X286, .([], .([], .([], .([], [])))), X287)\n\nT132 is ground\nX286 is free; \nX287 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, X286, .([], .([], .([], .([], [])))), X287), SCOPE: 7, CLAUSE: 3 | row2col(T132, T135, X286, .([], .([], .([], .([], [])))), X287), SCOPE: 7, CLAUSE: 4\n\nT132 is ground\nX286 is free; \nX287 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: row2col(T132, T135, X286, .([], .([], .([], .([], [])))), X287), SCOPE: 7, CLAUSE: 3\n\nT132 is ground\nX286 is free; \nX287 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: row2col(T132, T135, X286, .([], .([], .([], .([], [])))), X287), SCOPE: 7, CLAUSE: 4\n\nT132 is ground\nX286 is free; \nX287 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: row2col(T157, T160, X347, .([], .([], .([], .([], .([], []))))), X348)\n\nT157 is ground\nX347 is free; \nX348 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, X347, .([], .([], .([], .([], .([], []))))), X348), SCOPE: 8, CLAUSE: 3 | row2col(T157, T160, X347, .([], .([], .([], .([], .([], []))))), X348), SCOPE: 8, CLAUSE: 4\n\nT157 is ground\nX347 is free; \nX348 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: row2col(T157, T160, X347, .([], .([], .([], .([], .([], []))))), X348), SCOPE: 8, CLAUSE: 3\n\nT157 is ground\nX347 is free; \nX348 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: row2col(T157, T160, X347, .([], .([], .([], .([], .([], []))))), X348), SCOPE: 8, CLAUSE: 4\n\nT157 is ground\nX347 is free; \nX348 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: row2col(T182, T185, X408, .([], .([], .([], .([], .([], .([], [])))))), X409)\n\nT182 is ground\nX408 is free; \nX409 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, X408, .([], .([], .([], .([], .([], .([], [])))))), X409), SCOPE: 9, CLAUSE: 3 | row2col(T182, T185, X408, .([], .([], .([], .([], .([], .([], [])))))), X409), SCOPE: 9, CLAUSE: 4\n\nT182 is ground\nX408 is free; \nX409 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: row2col(T182, T185, X408, .([], .([], .([], .([], .([], .([], [])))))), X409), SCOPE: 9, CLAUSE: 3\n\nT182 is ground\nX408 is free; \nX409 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: row2col(T182, T185, X408, .([], .([], .([], .([], .([], .([], [])))))), X409), SCOPE: 9, CLAUSE: 4\n\nT182 is ground\nX408 is free; \nX409 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: row2col(T207, T210, X469, .([], .([], .([], .([], .([], .([], .([], []))))))), X470)\n\nT207 is ground\nX469 is free; \nX470 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, X469, .([], .([], .([], .([], .([], .([], .([], []))))))), X470), SCOPE: 10, CLAUSE: 3 | row2col(T207, T210, X469, .([], .([], .([], .([], .([], .([], .([], []))))))), X470), SCOPE: 10, CLAUSE: 4\n\nT207 is ground\nX469 is free; \nX470 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: row2col(T207, T210, X469, .([], .([], .([], .([], .([], .([], .([], []))))))), X470), SCOPE: 10, CLAUSE: 3\n\nT207 is ground\nX469 is free; \nX470 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: row2col(T207, T210, X469, .([], .([], .([], .([], .([], .([], .([], []))))))), X470), SCOPE: 10, CLAUSE: 4\n\nT207 is ground\nX469 is free; \nX470 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: row2col(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], .([], [])))))))), X531)\n\nT232 is ground\nX530 is free; \nX531 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: row2col(T232, T235, X530, .([], T240), X531)\n\nT232 is ground\nT240 is ground\nX530 is free; \nX531 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: row2col(T232, T235, X530, .([], T240), X531), SCOPE: 11, CLAUSE: 3 | row2col(T232, T235, X530, .([], T240), X531), SCOPE: 11, CLAUSE: 4\n\nT232 is ground\nT240 is ground\nX530 is free; \nX531 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: row2col(T232, T235, X530, .([], T240), X531), SCOPE: 11, CLAUSE: 3\n\nT232 is ground\nT240 is ground\nX530 is free; \nX531 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: row2col(T232, T235, X530, .([], T240), X531), SCOPE: 11, CLAUSE: 4\n\nT232 is ground\nT240 is ground\nX530 is free; \nX531 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: row2col(T267, T271, X600, .([], .([], T270)), X601)\n\nT267 is ground\nT270 is ground\nX600 is free; \nX601 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), X687, [], X688), transpose_aux(T306, X688, X687))\n\nT305 is ground\nT306 is ground\nX687 is free; \nX688 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 substitution\nT1 -> T5\nX4 -> T5\nT2 -> T7\nX5 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = white];
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 substitution\nX30 -> 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 = lightblue];
5 -> 99 [arrowhead = none ];
99 -> 8;

100 [label="EVAL with clause\ntranspose_aux([], X709, X709).\nand substitution\nT5 -> []\nX709 -> []\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 100 [arrowhead = none ];
100 -> 90;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT35 is ground\nreplacements:\nX35 -> 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(.(X96, X97), .(.(X96, X98), X99), .(X98, X100), X101, X102) :- row2col(X97, X99, X100, .([], X101), X102).\nand substitution\nX96 -> T56\nX97 -> T57\nT24 -> .(T56, T57)\nX98 -> T58\nT28 -> .(T56, T58)\nT29 -> T60\nX99 -> T60\nX100 -> X103\nX35 -> .(T58, X103)\nX101 -> []\nX36 -> X104\nX102 -> X104\nT59 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 108 [arrowhead = none ];
108 -> 14;

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

110 [label="BACKTRACK\nfor clause: row2col([], [], [], A, A)because of non-unification", fontsize=14, style = filled, fillcolor = white];
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(.(X157, X158), .(.(X157, X159), X160), .(X159, X161), X162, X163) :- row2col(X158, X160, X161, .([], X162), X163).\nand substitution\nX157 -> T81\nX158 -> T82\nT57 -> .(T81, T82)\nX159 -> T83\nX160 -> T85\nT60 -> .(.(T81, T83), T85)\nX161 -> X164\nX103 -> .(T83, X164)\nX162 -> .([], [])\nX104 -> X165\nX163 -> X165\nT84 -> T85", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 114 [arrowhead = none ];
114 -> 19;

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

116 [label="EVAL with clause\nrow2col([], [], [], X648, X648).\nand substitution\nT57 -> []\nT60 -> []\nX103 -> []\nX648 -> .([], [])\nX104 -> .([], [])", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 116 [arrowhead = none ];
116 -> 78;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X218, X219), .(.(X218, X220), X221), .(X220, X222), X223, X224) :- row2col(X219, X221, X222, .([], X223), X224).\nand substitution\nX218 -> T106\nX219 -> T107\nT82 -> .(T106, T107)\nX220 -> T108\nX221 -> T110\nT85 -> .(.(T106, T108), T110)\nX222 -> X225\nX164 -> .(T108, X225)\nX223 -> .([], .([], []))\nX165 -> X226\nX224 -> X226\nT109 -> T110", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 121 [arrowhead = none ];
121 -> 24;

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

123 [label="EVAL with clause\nrow2col([], [], [], X644, X644).\nand substitution\nT82 -> []\nT85 -> []\nX164 -> []\nX644 -> .([], .([], []))\nX165 -> .([], .([], []))", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 123 [arrowhead = none ];
123 -> 75;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X279, X280), .(.(X279, X281), X282), .(X281, X283), X284, X285) :- row2col(X280, X282, X283, .([], X284), X285).\nand substitution\nX279 -> T131\nX280 -> T132\nT107 -> .(T131, T132)\nX281 -> T133\nX282 -> T135\nT110 -> .(.(T131, T133), T135)\nX283 -> X286\nX225 -> .(T133, X286)\nX284 -> .([], .([], .([], [])))\nX226 -> X287\nX285 -> X287\nT134 -> T135", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 128 [arrowhead = none ];
128 -> 29;

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

130 [label="EVAL with clause\nrow2col([], [], [], X640, X640).\nand substitution\nT107 -> []\nT110 -> []\nX225 -> []\nX640 -> .([], .([], .([], [])))\nX226 -> .([], .([], .([], [])))", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 130 [arrowhead = none ];
130 -> 72;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X340, X341), .(.(X340, X342), X343), .(X342, X344), X345, X346) :- row2col(X341, X343, X344, .([], X345), X346).\nand substitution\nX340 -> T156\nX341 -> T157\nT132 -> .(T156, T157)\nX342 -> T158\nX343 -> T160\nT135 -> .(.(T156, T158), T160)\nX344 -> X347\nX286 -> .(T158, X347)\nX345 -> .([], .([], .([], .([], []))))\nX287 -> X348\nX346 -> X348\nT159 -> T160", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 135 [arrowhead = none ];
135 -> 34;

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

137 [label="EVAL with clause\nrow2col([], [], [], X636, X636).\nand substitution\nT132 -> []\nT135 -> []\nX286 -> []\nX636 -> .([], .([], .([], .([], []))))\nX287 -> .([], .([], .([], .([], []))))", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 137 [arrowhead = none ];
137 -> 69;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X401, X402), .(.(X401, X403), X404), .(X403, X405), X406, X407) :- row2col(X402, X404, X405, .([], X406), X407).\nand substitution\nX401 -> T181\nX402 -> T182\nT157 -> .(T181, T182)\nX403 -> T183\nX404 -> T185\nT160 -> .(.(T181, T183), T185)\nX405 -> X408\nX347 -> .(T183, X408)\nX406 -> .([], .([], .([], .([], .([], [])))))\nX348 -> X409\nX407 -> X409\nT184 -> T185", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 142 [arrowhead = none ];
142 -> 39;

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

144 [label="EVAL with clause\nrow2col([], [], [], X632, X632).\nand substitution\nT157 -> []\nT160 -> []\nX347 -> []\nX632 -> .([], .([], .([], .([], .([], [])))))\nX348 -> .([], .([], .([], .([], .([], [])))))", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 144 [arrowhead = none ];
144 -> 66;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X462, X463), .(.(X462, X464), X465), .(X464, X466), X467, X468) :- row2col(X463, X465, X466, .([], X467), X468).\nand substitution\nX462 -> T206\nX463 -> T207\nT182 -> .(T206, T207)\nX464 -> T208\nX465 -> T210\nT185 -> .(.(T206, T208), T210)\nX466 -> X469\nX408 -> .(T208, X469)\nX467 -> .([], .([], .([], .([], .([], .([], []))))))\nX409 -> X470\nX468 -> X470\nT209 -> T210", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 149 [arrowhead = none ];
149 -> 44;

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

151 [label="EVAL with clause\nrow2col([], [], [], X628, X628).\nand substitution\nT182 -> []\nT185 -> []\nX408 -> []\nX628 -> .([], .([], .([], .([], .([], .([], []))))))\nX409 -> .([], .([], .([], .([], .([], .([], []))))))", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 151 [arrowhead = none ];
151 -> 63;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X523, X524), .(.(X523, X525), X526), .(X525, X527), X528, X529) :- row2col(X524, X526, X527, .([], X528), X529).\nand substitution\nX523 -> T231\nX524 -> T232\nT207 -> .(T231, T232)\nX525 -> T233\nX526 -> T235\nT210 -> .(.(T231, T233), T235)\nX527 -> X530\nX469 -> .(T233, X530)\nX528 -> .([], .([], .([], .([], .([], .([], .([], [])))))))\nX470 -> X531\nX529 -> X531\nT234 -> T235", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 156 [arrowhead = none ];
156 -> 49;

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

158 [label="EVAL with clause\nrow2col([], [], [], X624, X624).\nand substitution\nT207 -> []\nT210 -> []\nX469 -> []\nX624 -> .([], .([], .([], .([], .([], .([], .([], [])))))))\nX470 -> .([], .([], .([], .([], .([], .([], .([], [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 158 [arrowhead = none ];
158 -> 60;

159 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(.(X593, X594), .(.(X593, X595), X596), .(X595, X597), X598, X599) :- row2col(X594, X596, X597, .([], X598), X599).\nand substitution\nX593 -> T266\nX594 -> T267\nT232 -> .(T266, T267)\nX595 -> T268\nX596 -> T271\nT235 -> .(.(T266, T268), T271)\nX597 -> X600\nX530 -> .(T268, X600)\nT240 -> T270\nX598 -> .([], T270)\nX531 -> X601\nX599 -> X601\nT269 -> T271", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 164 [arrowhead = none ];
164 -> 55;

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

166 [label="EVAL with clause\nrow2col([], [], [], X620, X620).\nand substitution\nT232 -> []\nT235 -> []\nX530 -> []\nT240 -> T279\nX620 -> .([], T279)\nX531 -> .([], T279)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 166 [arrowhead = none ];
166 -> 57;

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

168 [label="INSTANCE with matching:\nT232 -> T267\nT235 -> T271\nX530 -> X600\nT240 -> .([], T270)\nX531 -> X601", 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(.(X682, X683), X684, .(X685, X686)) :- ','(row2col(X682, .(X685, X686), X687, [], X688), transpose_aux(X683, X688, X687)).\nand substitution\nX682 -> T305\nX683 -> T306\nT25 -> .(T305, T306)\nT35 -> T307\nX684 -> T307\nX685 -> T310\nX686 -> 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 = lightblue];
83 -> 180 [arrowhead = none ];
180 -> 86;

181 [label="EVAL with clause\ntranspose_aux([], X705, X705).\nand substitution\nT25 -> []\nT35 -> T318\nX705 -> T318\nT34 -> T318", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 181 [arrowhead = none ];
181 -> 87;

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

183 [label="INSTANCE with matching:\nT24 -> T305\nT28 -> T310\nT29 -> T311\nX35 -> X687\nX36 -> X688\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:

Clauses:

row2col51(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) :- row2col51(T267, T271, X600, .([], T270), X601).
row2col51([], [], [], T279, .([], T279)).
p7(T24, T28, T29, X35, X36, T25) :- row2col9(T24, T28, T29, X35, X36).
p7(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) :- ','(row2col9(T24, T28, T29, .(T310, T311), T307), p7(T305, T310, T311, X687, X688, T306)).
p7(T24, T28, T29, T318, T318, []) :- row2col9(T24, T28, T29, T318, T318).
row2col9(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) :- row2col51(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531).
row2col9(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))).
row2col9(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], []))))))).
row2col9(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))).
row2col9(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))).
row2col9(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))).
row2col9(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))).
row2col9(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])).
transpose1(.(T24, T25), .(T28, T29)) :- p7(T24, T28, T29, X35, X36, T25).
transpose1([], []).

Queries:

transpose1(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:
transpose1_in: (b,f)
p7_in: (b,f,f,f,f,b)
row2col9_in: (b,f,f,f,f)
row2col51_in: (b,f,f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose1_in_ga(.(T24, T25), .(T28, T29)) → U7_ga(T24, T25, T28, T29, p7_in_gaaaag(T24, T28, T29, X35, X36, T25))
p7_in_gaaaag(T24, T28, T29, X35, X36, T25) → U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_in_gaaaa(T24, T28, T29, X35, X36))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_in_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531))
row2col51_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_in_gaaga(T267, T271, X600, .([], T270), X601))
row2col51_in_gaaga([], [], [], T279, .([], T279)) → row2col51_out_gaaga([], [], [], T279, .([], T279))
U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_out_gaaga(T267, T271, X600, .([], T270), X601)) → row2col51_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601)
U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_out_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531)) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531)
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2col9_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2col9_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2col9_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2col9_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_out_gaaaa(T24, T28, T29, X35, X36)) → p7_out_gaaaag(T24, T28, T29, X35, X36, T25)
p7_in_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_in_gaaaag(T305, T310, T311, X687, X688, T306))
p7_in_gaaaag(T24, T28, T29, T318, T318, []) → U5_gaaaag(T24, T28, T29, T318, row2col9_in_gaaaa(T24, T28, T29, T318, T318))
U5_gaaaag(T24, T28, T29, T318, row2col9_out_gaaaa(T24, T28, T29, T318, T318)) → p7_out_gaaaag(T24, T28, T29, T318, T318, [])
U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_out_gaaaag(T305, T310, T311, X687, X688, T306)) → p7_out_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306))
U7_ga(T24, T25, T28, T29, p7_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transpose1_out_ga(.(T24, T25), .(T28, T29))
transpose1_in_ga([], []) → transpose1_out_ga([], [])

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

transpose1_in_ga(.(T24, T25), .(T28, T29)) → U7_ga(T24, T25, T28, T29, p7_in_gaaaag(T24, T28, T29, X35, X36, T25))
p7_in_gaaaag(T24, T28, T29, X35, X36, T25) → U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_in_gaaaa(T24, T28, T29, X35, X36))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_in_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531))
row2col51_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_in_gaaga(T267, T271, X600, .([], T270), X601))
row2col51_in_gaaga([], [], [], T279, .([], T279)) → row2col51_out_gaaga([], [], [], T279, .([], T279))
U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_out_gaaga(T267, T271, X600, .([], T270), X601)) → row2col51_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601)
U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_out_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531)) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531)
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2col9_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2col9_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2col9_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2col9_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_out_gaaaa(T24, T28, T29, X35, X36)) → p7_out_gaaaag(T24, T28, T29, X35, X36, T25)
p7_in_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_in_gaaaag(T305, T310, T311, X687, X688, T306))
p7_in_gaaaag(T24, T28, T29, T318, T318, []) → U5_gaaaag(T24, T28, T29, T318, row2col9_in_gaaaa(T24, T28, T29, T318, T318))
U5_gaaaag(T24, T28, T29, T318, row2col9_out_gaaaa(T24, T28, T29, T318, T318)) → p7_out_gaaaag(T24, T28, T29, T318, T318, [])
U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_out_gaaaag(T305, T310, T311, X687, X688, T306)) → p7_out_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306))
U7_ga(T24, T25, T28, T29, p7_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transpose1_out_ga(.(T24, T25), .(T28, T29))
transpose1_in_ga([], []) → transpose1_out_ga([], [])

(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:

TRANSPOSE1_IN_GA(.(T24, T25), .(T28, T29)) → U7_GA(T24, T25, T28, T29, p7_in_gaaaag(T24, T28, T29, X35, X36, T25))
TRANSPOSE1_IN_GA(.(T24, T25), .(T28, T29)) → P7_IN_GAAAAG(T24, T28, T29, X35, X36, T25)
P7_IN_GAAAAG(T24, T28, T29, X35, X36, T25) → U2_GAAAAG(T24, T28, T29, X35, X36, T25, row2col9_in_gaaaa(T24, T28, T29, X35, X36))
P7_IN_GAAAAG(T24, T28, T29, X35, X36, T25) → ROW2COL9_IN_GAAAA(T24, T28, T29, X35, X36)
ROW2COL9_IN_GAAAA(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → U6_GAAAA(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_in_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531))
ROW2COL9_IN_GAAAA(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → ROW2COL51_IN_GAAGA(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531)
ROW2COL51_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → U1_GAAGA(T266, T267, T268, T271, X600, T270, X601, row2col51_in_gaaga(T267, T271, X600, .([], T270), X601))
ROW2COL51_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → ROW2COL51_IN_GAAGA(T267, T271, X600, .([], T270), X601)
P7_IN_GAAAAG(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
P7_IN_GAAAAG(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → ROW2COL9_IN_GAAAA(T24, T28, T29, .(T310, T311), T307)
U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → U4_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, p7_in_gaaaag(T305, T310, T311, X687, X688, T306))
U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → P7_IN_GAAAAG(T305, T310, T311, X687, X688, T306)
P7_IN_GAAAAG(T24, T28, T29, T318, T318, []) → U5_GAAAAG(T24, T28, T29, T318, row2col9_in_gaaaa(T24, T28, T29, T318, T318))
P7_IN_GAAAAG(T24, T28, T29, T318, T318, []) → ROW2COL9_IN_GAAAA(T24, T28, T29, T318, T318)

The TRS R consists of the following rules:

transpose1_in_ga(.(T24, T25), .(T28, T29)) → U7_ga(T24, T25, T28, T29, p7_in_gaaaag(T24, T28, T29, X35, X36, T25))
p7_in_gaaaag(T24, T28, T29, X35, X36, T25) → U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_in_gaaaa(T24, T28, T29, X35, X36))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_in_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531))
row2col51_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_in_gaaga(T267, T271, X600, .([], T270), X601))
row2col51_in_gaaga([], [], [], T279, .([], T279)) → row2col51_out_gaaga([], [], [], T279, .([], T279))
U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_out_gaaga(T267, T271, X600, .([], T270), X601)) → row2col51_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601)
U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_out_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531)) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531)
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2col9_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2col9_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2col9_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2col9_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_out_gaaaa(T24, T28, T29, X35, X36)) → p7_out_gaaaag(T24, T28, T29, X35, X36, T25)
p7_in_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_in_gaaaag(T305, T310, T311, X687, X688, T306))
p7_in_gaaaag(T24, T28, T29, T318, T318, []) → U5_gaaaag(T24, T28, T29, T318, row2col9_in_gaaaa(T24, T28, T29, T318, T318))
U5_gaaaag(T24, T28, T29, T318, row2col9_out_gaaaa(T24, T28, T29, T318, T318)) → p7_out_gaaaag(T24, T28, T29, T318, T318, [])
U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_out_gaaaag(T305, T310, T311, X687, X688, T306)) → p7_out_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306))
U7_ga(T24, T25, T28, T29, p7_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transpose1_out_ga(.(T24, T25), .(T28, T29))
transpose1_in_ga([], []) → transpose1_out_ga([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_ga(x1, x2) = transpose1_in_ga(x1)
.(x1, x2) = .(x1, x2)
U7_ga(x1, x2, x3, x4, x5) = U7_ga(x5)
p7_in_gaaaag(x1, x2, x3, x4, x5, x6) = p7_in_gaaaag(x1, x6)
U2_gaaaag(x1, x2, x3, x4, x5, x6, x7) = U2_gaaaag(x7)
row2col9_in_gaaaa(x1, x2, x3, x4, x5) = row2col9_in_gaaaa(x1)
U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U6_gaaaa(x21)
row2col51_in_gaaga(x1, x2, x3, x4, x5) = row2col51_in_gaaga(x1, x4)
U1_gaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_gaaga(x8)
[] = []
row2col51_out_gaaga(x1, x2, x3, x4, x5) = row2col51_out_gaaga(x5)
row2col9_out_gaaaa(x1, x2, x3, x4, x5) = row2col9_out_gaaaa(x5)
p7_out_gaaaag(x1, x2, x3, x4, x5, x6) = p7_out_gaaaag(x5)
U3_gaaaag(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_gaaaag(x7, x8, x9)
U4_gaaaag(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_gaaaag(x6, x9)
U5_gaaaag(x1, x2, x3, x4, x5) = U5_gaaaag(x5)
transpose1_out_ga(x1, x2) = transpose1_out_ga
TRANSPOSE1_IN_GA(x1, x2) = TRANSPOSE1_IN_GA(x1)
U7_GA(x1, x2, x3, x4, x5) = U7_GA(x5)
P7_IN_GAAAAG(x1, x2, x3, x4, x5, x6) = P7_IN_GAAAAG(x1, x6)
U2_GAAAAG(x1, x2, x3, x4, x5, x6, x7) = U2_GAAAAG(x7)
ROW2COL9_IN_GAAAA(x1, x2, x3, x4, x5) = ROW2COL9_IN_GAAAA(x1)
U6_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U6_GAAAA(x21)
ROW2COL51_IN_GAAGA(x1, x2, x3, x4, x5) = ROW2COL51_IN_GAAGA(x1, x4)
U1_GAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_GAAGA(x8)
U3_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_GAAAAG(x7, x8, x9)
U4_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U4_GAAAAG(x6, x9)
U5_GAAAAG(x1, x2, x3, x4, x5) = U5_GAAAAG(x5)

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

(6) Obligation:

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

TRANSPOSE1_IN_GA(.(T24, T25), .(T28, T29)) → U7_GA(T24, T25, T28, T29, p7_in_gaaaag(T24, T28, T29, X35, X36, T25))
TRANSPOSE1_IN_GA(.(T24, T25), .(T28, T29)) → P7_IN_GAAAAG(T24, T28, T29, X35, X36, T25)
P7_IN_GAAAAG(T24, T28, T29, X35, X36, T25) → U2_GAAAAG(T24, T28, T29, X35, X36, T25, row2col9_in_gaaaa(T24, T28, T29, X35, X36))
P7_IN_GAAAAG(T24, T28, T29, X35, X36, T25) → ROW2COL9_IN_GAAAA(T24, T28, T29, X35, X36)
ROW2COL9_IN_GAAAA(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → U6_GAAAA(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_in_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531))
ROW2COL9_IN_GAAAA(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → ROW2COL51_IN_GAAGA(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531)
ROW2COL51_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → U1_GAAGA(T266, T267, T268, T271, X600, T270, X601, row2col51_in_gaaga(T267, T271, X600, .([], T270), X601))
ROW2COL51_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → ROW2COL51_IN_GAAGA(T267, T271, X600, .([], T270), X601)
P7_IN_GAAAAG(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
P7_IN_GAAAAG(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → ROW2COL9_IN_GAAAA(T24, T28, T29, .(T310, T311), T307)
U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → U4_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, p7_in_gaaaag(T305, T310, T311, X687, X688, T306))
U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → P7_IN_GAAAAG(T305, T310, T311, X687, X688, T306)
P7_IN_GAAAAG(T24, T28, T29, T318, T318, []) → U5_GAAAAG(T24, T28, T29, T318, row2col9_in_gaaaa(T24, T28, T29, T318, T318))
P7_IN_GAAAAG(T24, T28, T29, T318, T318, []) → ROW2COL9_IN_GAAAA(T24, T28, T29, T318, T318)

The TRS R consists of the following rules:

transpose1_in_ga(.(T24, T25), .(T28, T29)) → U7_ga(T24, T25, T28, T29, p7_in_gaaaag(T24, T28, T29, X35, X36, T25))
p7_in_gaaaag(T24, T28, T29, X35, X36, T25) → U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_in_gaaaa(T24, T28, T29, X35, X36))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_in_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531))
row2col51_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_in_gaaga(T267, T271, X600, .([], T270), X601))
row2col51_in_gaaga([], [], [], T279, .([], T279)) → row2col51_out_gaaga([], [], [], T279, .([], T279))
U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_out_gaaga(T267, T271, X600, .([], T270), X601)) → row2col51_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601)
U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_out_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531)) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531)
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2col9_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2col9_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2col9_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2col9_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_out_gaaaa(T24, T28, T29, X35, X36)) → p7_out_gaaaag(T24, T28, T29, X35, X36, T25)
p7_in_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_in_gaaaag(T305, T310, T311, X687, X688, T306))
p7_in_gaaaag(T24, T28, T29, T318, T318, []) → U5_gaaaag(T24, T28, T29, T318, row2col9_in_gaaaa(T24, T28, T29, T318, T318))
U5_gaaaag(T24, T28, T29, T318, row2col9_out_gaaaa(T24, T28, T29, T318, T318)) → p7_out_gaaaag(T24, T28, T29, T318, T318, [])
U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_out_gaaaag(T305, T310, T311, X687, X688, T306)) → p7_out_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306))
U7_ga(T24, T25, T28, T29, p7_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transpose1_out_ga(.(T24, T25), .(T28, T29))
transpose1_in_ga([], []) → transpose1_out_ga([], [])

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 11 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ROW2COL51_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → ROW2COL51_IN_GAAGA(T267, T271, X600, .([], T270), X601)

The TRS R consists of the following rules:

transpose1_in_ga(.(T24, T25), .(T28, T29)) → U7_ga(T24, T25, T28, T29, p7_in_gaaaag(T24, T28, T29, X35, X36, T25))
p7_in_gaaaag(T24, T28, T29, X35, X36, T25) → U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_in_gaaaa(T24, T28, T29, X35, X36))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_in_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531))
row2col51_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_in_gaaga(T267, T271, X600, .([], T270), X601))
row2col51_in_gaaga([], [], [], T279, .([], T279)) → row2col51_out_gaaga([], [], [], T279, .([], T279))
U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_out_gaaga(T267, T271, X600, .([], T270), X601)) → row2col51_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601)
U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_out_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531)) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531)
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2col9_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2col9_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2col9_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2col9_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_out_gaaaa(T24, T28, T29, X35, X36)) → p7_out_gaaaag(T24, T28, T29, X35, X36, T25)
p7_in_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_in_gaaaag(T305, T310, T311, X687, X688, T306))
p7_in_gaaaag(T24, T28, T29, T318, T318, []) → U5_gaaaag(T24, T28, T29, T318, row2col9_in_gaaaa(T24, T28, T29, T318, T318))
U5_gaaaag(T24, T28, T29, T318, row2col9_out_gaaaa(T24, T28, T29, T318, T318)) → p7_out_gaaaag(T24, T28, T29, T318, T318, [])
U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_out_gaaaag(T305, T310, T311, X687, X688, T306)) → p7_out_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306))
U7_ga(T24, T25, T28, T29, p7_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transpose1_out_ga(.(T24, T25), .(T28, T29))
transpose1_in_ga([], []) → transpose1_out_ga([], [])

(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:

ROW2COL51_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → ROW2COL51_IN_GAAGA(T267, T271, X600, .([], T270), X601)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COL51_IN_GAAGA(x1, x2, x3, x4, x5) = ROW2COL51_IN_GAAGA(x1, x4)

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:

ROW2COL51_IN_GAAGA(.(T266, T267), T270) → ROW2COL51_IN_GAAGA(T267, .([], T270))

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

(14) 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:

ROW2COL51_IN_GAAGA(.(T266, T267), T270) → ROW2COL51_IN_GAAGA(T267, .([], T270))
The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

P7_IN_GAAAAG(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → P7_IN_GAAAAG(T305, T310, T311, X687, X688, T306)

The TRS R consists of the following rules:

transpose1_in_ga(.(T24, T25), .(T28, T29)) → U7_ga(T24, T25, T28, T29, p7_in_gaaaag(T24, T28, T29, X35, X36, T25))
p7_in_gaaaag(T24, T28, T29, X35, X36, T25) → U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_in_gaaaa(T24, T28, T29, X35, X36))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_in_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531))
row2col51_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_in_gaaga(T267, T271, X600, .([], T270), X601))
row2col51_in_gaaga([], [], [], T279, .([], T279)) → row2col51_out_gaaga([], [], [], T279, .([], T279))
U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_out_gaaga(T267, T271, X600, .([], T270), X601)) → row2col51_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601)
U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_out_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531)) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531)
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2col9_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2col9_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2col9_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2col9_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U2_gaaaag(T24, T28, T29, X35, X36, T25, row2col9_out_gaaaa(T24, T28, T29, X35, X36)) → p7_out_gaaaag(T24, T28, T29, X35, X36, T25)
p7_in_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
U3_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_in_gaaaag(T305, T310, T311, X687, X688, T306))
p7_in_gaaaag(T24, T28, T29, T318, T318, []) → U5_gaaaag(T24, T28, T29, T318, row2col9_in_gaaaa(T24, T28, T29, T318, T318))
U5_gaaaag(T24, T28, T29, T318, row2col9_out_gaaaa(T24, T28, T29, T318, T318)) → p7_out_gaaaag(T24, T28, T29, T318, T318, [])
U4_gaaaag(T24, T28, T29, T310, T311, T307, T305, T306, p7_out_gaaaag(T305, T310, T311, X687, X688, T306)) → p7_out_gaaaag(T24, T28, T29, .(T310, T311), T307, .(T305, T306))
U7_ga(T24, T25, T28, T29, p7_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transpose1_out_ga(.(T24, T25), .(T28, T29))
transpose1_in_ga([], []) → transpose1_out_ga([], [])

(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:

P7_IN_GAAAAG(T24, T28, T29, .(T310, T311), T307, .(T305, T306)) → U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_in_gaaaa(T24, T28, T29, .(T310, T311), T307))
U3_GAAAAG(T24, T28, T29, T310, T311, T307, T305, T306, row2col9_out_gaaaa(T24, T28, T29, .(T310, T311), T307)) → P7_IN_GAAAAG(T305, T310, T311, X687, X688, T306)

The TRS R consists of the following rules:

row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531) → U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_in_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), [])))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), []))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2col9_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2col9_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2col9_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2col9_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U6_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X530, X531, row2col51_out_gaaga(T232, T235, X530, .([], .([], .([], .([], .([], .([], .([], []))))))), X531)) → row2col9_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), .(.(T181, T183), .(.(T206, T208), .(.(T231, T233), T235))))))), .(T58, .(T83, .(T108, .(T133, .(T158, .(T183, .(T208, .(T233, X530)))))))), X531)
row2col51_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601) → U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_in_gaaga(T267, T271, X600, .([], T270), X601))
row2col51_in_gaaga([], [], [], T279, .([], T279)) → row2col51_out_gaaga([], [], [], T279, .([], T279))
U1_gaaga(T266, T267, T268, T271, X600, T270, X601, row2col51_out_gaaga(T267, T271, X600, .([], T270), X601)) → row2col51_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X600), T270, X601)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
row2col9_in_gaaaa(x1, x2, x3, x4, x5) = row2col9_in_gaaaa(x1)
U6_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U6_gaaaa(x21)
row2col51_in_gaaga(x1, x2, x3, x4, x5) = row2col51_in_gaaga(x1, x4)
U1_gaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_gaaga(x8)
[] = []
row2col51_out_gaaga(x1, x2, x3, x4, x5) = row2col51_out_gaaga(x5)
row2col9_out_gaaaa(x1, x2, x3, x4, x5) = row2col9_out_gaaaa(x5)
P7_IN_GAAAAG(x1, x2, x3, x4, x5, x6) = P7_IN_GAAAAG(x1, x6)
U3_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U3_GAAAAG(x7, x8, x9)

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:

P7_IN_GAAAAG(T24, .(T305, T306)) → U3_GAAAAG(T305, T306, row2col9_in_gaaaa(T24))
U3_GAAAAG(T305, T306, row2col9_out_gaaaa(T307)) → P7_IN_GAAAAG(T305, T306)

The TRS R consists of the following rules:

row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232))))))))) → U6_gaaaa(row2col51_in_gaaga(T232, .([], .([], .([], .([], .([], .([], .([], [])))))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, [])))))))) → row2col9_out_gaaaa(.([], .([], .([], .([], .([], .([], .([], []))))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, []))))))) → row2col9_out_gaaaa(.([], .([], .([], .([], .([], .([], [])))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, [])))))) → row2col9_out_gaaaa(.([], .([], .([], .([], .([], []))))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, .(T131, []))))) → row2col9_out_gaaaa(.([], .([], .([], .([], [])))))
row2col9_in_gaaaa(.(T56, .(T81, .(T106, [])))) → row2col9_out_gaaaa(.([], .([], .([], []))))
row2col9_in_gaaaa(.(T56, .(T81, []))) → row2col9_out_gaaaa(.([], .([], [])))
row2col9_in_gaaaa(.(T56, [])) → row2col9_out_gaaaa(.([], []))
U6_gaaaa(row2col51_out_gaaga(X531)) → row2col9_out_gaaaa(X531)
row2col51_in_gaaga(.(T266, T267), T270) → U1_gaaga(row2col51_in_gaaga(T267, .([], T270)))
row2col51_in_gaaga([], T279) → row2col51_out_gaaga(.([], T279))
U1_gaaga(row2col51_out_gaaga(X601)) → row2col51_out_gaaga(X601)

The set Q consists of the following terms:

row2col9_in_gaaaa(x0)
U6_gaaaa(x0)
row2col51_in_gaaga(x0, x1)
U1_gaaga(x0)

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

(21) 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(T305, T306, row2col9_out_gaaaa(T307)) → P7_IN_GAAAAG(T305, T306)
The graph contains the following edges 1 >= 1, 2 >= 2
P7_IN_GAAAAG(T24, .(T305, T306)) → U3_GAAAAG(T305, T306, row2col9_in_gaaaa(T24))
The graph contains the following edges 2 > 1, 2 > 2

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES