Left Termination proof of /tmp/tmpfUwnq

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 466 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 87 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 9 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 UsableRulesProof (⇔, 0 ms)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇒, 21 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: transpose(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: transpose(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: transpose_aux(T5, [], T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: 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];
12 [label="12: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [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];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="D: row2col(T24, .(T28, T29), X35, [], X36)\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="E: transpose_aux(T25, T35, T34)\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: 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];
24 [label="24: 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];
25 [label="25: 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];
32 [label="32: row2col(T57, T60, X98, .([], []), X99)\n\nT57 is ground\nX98 is free\nX99 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: 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];
39 [label="39: 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];
40 [label="40: 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];
45 [label="45: row2col(T82, T85, X152, .([], .([], [])), X153)\n\nT82 is ground\nX152 is free\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: 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];
54 [label="54: 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];
55 [label="55: 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];
60 [label="60: row2col(T107, T110, X206, .([], .([], .([], []))), X207)\n\nT107 is ground\nX206 is free\nX207 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: 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];
68 [label="68: 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];
69 [label="69: 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];
75 [label="75: row2col(T132, T135, X260, .([], .([], .([], .([], [])))), X261)\n\nT132 is ground\nX260 is free\nX261 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: 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];
83 [label="83: 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];
84 [label="84: 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];
87 [label="87: row2col(T157, T160, X314, .([], .([], .([], .([], .([], []))))), X315)\n\nT157 is ground\nX314 is free\nX315 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: 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];
95 [label="95: 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];
96 [label="96: 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];
101 [label="101: row2col(T182, T185, X368, .([], .([], .([], .([], .([], .([], [])))))), X369)\n\nT182 is ground\nX368 is free\nX369 is free", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: 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];
108 [label="108: 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];
109 [label="109: 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];
114 [label="114: row2col(T207, T210, X422, .([], .([], .([], .([], .([], .([], .([], []))))))), X423)\n\nT207 is ground\nX422 is free\nX423 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: 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];
121 [label="121: 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];
122 [label="122: 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];
128 [label="128: row2col(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], .([], [])))))))), X477)\n\nT232 is ground\nX476 is free\nX477 is free", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="C: 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];
135 [label="135: 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];
139 [label="139: 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];
140 [label="140: 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];
144 [label="144: 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];
145 [label="145: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="158: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
165 [label="165: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
168 [label="168: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
178 [label="178: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
181 [label="181: 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];
182 [label="182: transpose_aux(T25, T35, T34), SCOPE: 12, CLAUSE: 1\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="183: transpose_aux(T25, T35, T34), SCOPE: 12, CLAUSE: 2\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
187 [label="187: ','(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];
188 [label="188: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="191: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
193 [label="193: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
194 [label="194: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
195 [label="195: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
196 [label="196: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 197 [arrowhead = none ];
197 -> 5;

198 [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];
5 -> 198 [arrowhead = none ];
198 -> 7;

199 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 199 [arrowhead = none ];
199 -> 8;

200 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 200 [arrowhead = none ];
200 -> 12;

201 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 201 [arrowhead = none ];
201 -> 13;

202 [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];
12 -> 202 [arrowhead = none ];
202 -> 16;

203 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 203 [arrowhead = none ];
203 -> 17;

204 [label="EVAL with clause\ntranspose_aux([], X627, X627).\nand substitutionT5 -> [],\nX627 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 204 [arrowhead = none ];
204 -> 194;

205 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 205 [arrowhead = none ];
205 -> 195;

206 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 206 [arrowhead = none ];
206 -> 19;

207 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT35 is ground\nreplacements:X35 -> T34,\nX36 -> T35", fontsize=14, style = filled, fillcolor = violet];
16 -> 207 [arrowhead = none ];
207 -> 20;

208 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 208 [arrowhead = none ];
208 -> 23;

209 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 209 [arrowhead = none ];
209 -> 181;

210 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 210 [arrowhead = none ];
210 -> 24;

211 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 211 [arrowhead = none ];
211 -> 25;

212 [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];
24 -> 212 [arrowhead = none ];
212 -> 32;

213 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 213 [arrowhead = none ];
213 -> 33;

214 [label="BACKTRACK\nfor clause: row2col([], [], [], A, A)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 214 [arrowhead = none ];
214 -> 178;

215 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 215 [arrowhead = none ];
215 -> 37;

216 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 216 [arrowhead = none ];
216 -> 39;

217 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 217 [arrowhead = none ];
217 -> 40;

218 [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];
39 -> 218 [arrowhead = none ];
218 -> 45;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 219 [arrowhead = none ];
219 -> 46;

220 [label="EVAL with clause\nrow2col([], [], [], X573, X573).\nand substitutionT57 -> [],\nT60 -> [],\nX98 -> [],\nX573 -> .([], []),\nX99 -> .([], [])", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 220 [arrowhead = none ];
220 -> 175;

221 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 221 [arrowhead = none ];
221 -> 176;

222 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 222 [arrowhead = none ];
222 -> 51;

223 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 223 [arrowhead = none ];
223 -> 54;

224 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 224 [arrowhead = none ];
224 -> 55;

225 [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];
54 -> 225 [arrowhead = none ];
225 -> 60;

226 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 226 [arrowhead = none ];
226 -> 61;

227 [label="EVAL with clause\nrow2col([], [], [], X570, X570).\nand substitutionT82 -> [],\nT85 -> [],\nX152 -> [],\nX570 -> .([], .([], [])),\nX153 -> .([], .([], []))", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 227 [arrowhead = none ];
227 -> 172;

228 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 228 [arrowhead = none ];
228 -> 173;

229 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
60 -> 229 [arrowhead = none ];
229 -> 64;

230 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 230 [arrowhead = none ];
230 -> 68;

231 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 231 [arrowhead = none ];
231 -> 69;

232 [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];
68 -> 232 [arrowhead = none ];
232 -> 75;

233 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 233 [arrowhead = none ];
233 -> 76;

234 [label="EVAL with clause\nrow2col([], [], [], X567, X567).\nand substitutionT107 -> [],\nT110 -> [],\nX206 -> [],\nX567 -> .([], .([], .([], []))),\nX207 -> .([], .([], .([], [])))", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 234 [arrowhead = none ];
234 -> 168;

235 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 235 [arrowhead = none ];
235 -> 169;

236 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
75 -> 236 [arrowhead = none ];
236 -> 78;

237 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
78 -> 237 [arrowhead = none ];
237 -> 83;

238 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
78 -> 238 [arrowhead = none ];
238 -> 84;

239 [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];
83 -> 239 [arrowhead = none ];
239 -> 87;

240 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 240 [arrowhead = none ];
240 -> 88;

241 [label="EVAL with clause\nrow2col([], [], [], X564, X564).\nand substitutionT132 -> [],\nT135 -> [],\nX260 -> [],\nX564 -> .([], .([], .([], .([], [])))),\nX261 -> .([], .([], .([], .([], []))))", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 241 [arrowhead = none ];
241 -> 165;

242 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 242 [arrowhead = none ];
242 -> 166;

243 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
87 -> 243 [arrowhead = none ];
243 -> 92;

244 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
92 -> 244 [arrowhead = none ];
244 -> 95;

245 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
92 -> 245 [arrowhead = none ];
245 -> 96;

246 [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];
95 -> 246 [arrowhead = none ];
246 -> 101;

247 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 247 [arrowhead = none ];
247 -> 102;

248 [label="EVAL with clause\nrow2col([], [], [], X561, X561).\nand substitutionT157 -> [],\nT160 -> [],\nX314 -> [],\nX561 -> .([], .([], .([], .([], .([], []))))),\nX315 -> .([], .([], .([], .([], .([], [])))))", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 248 [arrowhead = none ];
248 -> 160;

249 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 249 [arrowhead = none ];
249 -> 161;

250 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
101 -> 250 [arrowhead = none ];
250 -> 106;

251 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 251 [arrowhead = none ];
251 -> 108;

252 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 252 [arrowhead = none ];
252 -> 109;

253 [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];
108 -> 253 [arrowhead = none ];
253 -> 114;

254 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 254 [arrowhead = none ];
254 -> 116;

255 [label="EVAL with clause\nrow2col([], [], [], X558, X558).\nand substitutionT182 -> [],\nT185 -> [],\nX368 -> [],\nX558 -> .([], .([], .([], .([], .([], .([], [])))))),\nX369 -> .([], .([], .([], .([], .([], .([], []))))))", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 255 [arrowhead = none ];
255 -> 156;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 256 [arrowhead = none ];
256 -> 157;

257 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
114 -> 257 [arrowhead = none ];
257 -> 119;

258 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
119 -> 258 [arrowhead = none ];
258 -> 121;

259 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
119 -> 259 [arrowhead = none ];
259 -> 122;

260 [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];
121 -> 260 [arrowhead = none ];
260 -> 128;

261 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
121 -> 261 [arrowhead = none ];
261 -> 129;

262 [label="EVAL with clause\nrow2col([], [], [], X555, X555).\nand substitutionT207 -> [],\nT210 -> [],\nX422 -> [],\nX555 -> .([], .([], .([], .([], .([], .([], .([], []))))))),\nX423 -> .([], .([], .([], .([], .([], .([], .([], [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
122 -> 262 [arrowhead = none ];
262 -> 153;

263 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
122 -> 263 [arrowhead = none ];
263 -> 154;

264 [label="GENERALIZATION\nT240 <-- .([], .([], .([], .([], .([], .([], .([], [])))))))\n\nNew Knowledge:\nT240 is ground", fontsize=14, style = filled, fillcolor = violet];
128 -> 264 [arrowhead = none ];
264 -> 132;

265 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
132 -> 265 [arrowhead = none ];
265 -> 135;

266 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
135 -> 266 [arrowhead = none ];
266 -> 139;

267 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
135 -> 267 [arrowhead = none ];
267 -> 140;

268 [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];
139 -> 268 [arrowhead = none ];
268 -> 144;

269 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
139 -> 269 [arrowhead = none ];
269 -> 145;

270 [label="EVAL with clause\nrow2col([], [], [], X552, X552).\nand substitutionT232 -> [],\nT235 -> [],\nX476 -> [],\nT240 -> T279,\nX552 -> .([], T279),\nX477 -> .([], T279)", fontsize=14, style = filled, fillcolor = lightblue];
140 -> 270 [arrowhead = none ];
270 -> 146;

271 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
140 -> 271 [arrowhead = none ];
271 -> 147;

272 [label="INSTANCE with matching:\nT232 -> T267\nT235 -> T271\nX476 -> X539\nT240 -> .([], T270)\nX477 -> X540", fontsize=14, style = filled, fillcolor = lightgrey];
144 -> 272 [arrowhead = none , style=dashed];
272 -> 132[style=dashed];

273 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
146 -> 273 [arrowhead = none ];
273 -> 148;

274 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
153 -> 274 [arrowhead = none ];
274 -> 155;

275 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
156 -> 275 [arrowhead = none ];
275 -> 158;

276 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
160 -> 276 [arrowhead = none ];
276 -> 162;

277 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
165 -> 277 [arrowhead = none ];
277 -> 167;

278 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
168 -> 278 [arrowhead = none ];
278 -> 170;

279 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
172 -> 279 [arrowhead = none ];
279 -> 174;

280 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
175 -> 280 [arrowhead = none ];
280 -> 177;

281 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
181 -> 281 [arrowhead = none ];
281 -> 182;

282 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
181 -> 282 [arrowhead = none ];
282 -> 183;

283 [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];
182 -> 283 [arrowhead = none ];
283 -> 187;

284 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
182 -> 284 [arrowhead = none ];
284 -> 188;

285 [label="EVAL with clause\ntranspose_aux([], X624, X624).\nand substitutionT25 -> [],\nT35 -> T318,\nX624 -> T318,\nT34 -> T318", fontsize=14, style = filled, fillcolor = lightblue];
183 -> 285 [arrowhead = none ];
285 -> 191;

286 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
183 -> 286 [arrowhead = none ];
286 -> 192;

287 [label="INSTANCE with matching:\nT24 -> T305\nT28 -> T310\nT29 -> T311\nX35 -> X611\nX36 -> X612\nT25 -> T306", fontsize=14, style = filled, fillcolor = lightgrey];
187 -> 287 [arrowhead = none , style=dashed];
287 -> 16[style=dashed];

288 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
191 -> 288 [arrowhead = none ];
288 -> 193;

289 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
194 -> 289 [arrowhead = none ];
289 -> 196;

}

(2) Obligation:

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

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_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, X476)))))))), X477) → U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_in_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477))
row2colC_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_in_gaaga(T267, T271, X539, .([], T270), X540))
row2colC_in_gaaga([], [], [], T279, .([], T279)) → row2colC_out_gaaga([], [], [], T279, .([], T279))
U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_out_gaaga(T267, T271, X539, .([], T270), X540)) → row2colC_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540)
U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_out_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477)) → row2colD_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, X476)))))))), X477)
row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2colD_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2colD_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2colD_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2colD_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T305, T306), T307, .(T310, T311)) → U4_gga(T305, T306, T307, T310, T311, pB_in_gaaaag(T305, T310, T311, X611, X612, T306))
U4_gga(T305, T306, T307, T310, T311, pB_out_gaaaag(T305, T310, T311, X611, X612, T306)) → transpose_auxE_out_gga(.(T305, T306), T307, .(T310, T311))
transpose_auxE_in_gga([], T318, T318) → transpose_auxE_out_gga([], T318, T318)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

(3) 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:

TRANSPOSEA_IN_GA(.(T24, T25), .(T28, T29)) → U1_GA(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
TRANSPOSEA_IN_GA(.(T24, T25), .(T28, T29)) → PB_IN_GAAAAG(T24, T28, T29, X35, X36, T25)
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → ROW2COLD_IN_GAAAA(T24, T28, T29, T34, T35)
ROW2COLD_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, X476)))))))), X477) → U3_GAAAA(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_in_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477))
ROW2COLD_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, X476)))))))), X477) → ROW2COLC_IN_GAAGA(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477)
ROW2COLC_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → U2_GAAGA(T266, T267, T268, T271, X539, T270, X540, row2colC_in_gaaga(T267, T271, X539, .([], T270), X540))
ROW2COLC_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → ROW2COLC_IN_GAAGA(T267, T271, X539, .([], T270), X540)
U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_GAAAAG(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35, T34)
TRANSPOSE_AUXE_IN_GGA(.(T305, T306), T307, .(T310, T311)) → U4_GGA(T305, T306, T307, T310, T311, pB_in_gaaaag(T305, T310, T311, X611, X612, T306))
TRANSPOSE_AUXE_IN_GGA(.(T305, T306), T307, .(T310, T311)) → PB_IN_GAAAAG(T305, T310, T311, X611, X612, T306)

The TRS R consists of the following rules:

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_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, X476)))))))), X477) → U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_in_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477))
row2colC_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_in_gaaga(T267, T271, X539, .([], T270), X540))
row2colC_in_gaaga([], [], [], T279, .([], T279)) → row2colC_out_gaaga([], [], [], T279, .([], T279))
U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_out_gaaga(T267, T271, X539, .([], T270), X540)) → row2colC_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540)
U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_out_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477)) → row2colD_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, X476)))))))), X477)
row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2colD_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2colD_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2colD_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2colD_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T305, T306), T307, .(T310, T311)) → U4_gga(T305, T306, T307, T310, T311, pB_in_gaaaag(T305, T310, T311, X611, X612, T306))
U4_gga(T305, T306, T307, T310, T311, pB_out_gaaaag(T305, T310, T311, X611, X612, T306)) → transpose_auxE_out_gga(.(T305, T306), T307, .(T310, T311))
transpose_auxE_in_gga([], T318, T318) → transpose_auxE_out_gga([], T318, T318)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_ga(x1, x2) = transposeA_in_ga(x1)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6) = pB_in_gaaaag(x1, x6)
U5_gaaaag(x1, x2, x3, x4, x5, x6, x7) = U5_gaaaag(x1, x6, x7)
row2colD_in_gaaaa(x1, x2, x3, x4, x5) = row2colD_in_gaaaa(x1)
U3_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U3_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x21)
row2colC_in_gaaga(x1, x2, x3, x4, x5) = row2colC_in_gaaga(x1, x4)
U2_gaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_gaaga(x1, x2, x6, x8)
[] = []
row2colC_out_gaaga(x1, x2, x3, x4, x5) = row2colC_out_gaaga(x1, x4, x5)
row2colD_out_gaaaa(x1, x2, x3, x4, x5) = row2colD_out_gaaaa(x1, x5)
U6_gaaaag(x1, x2, x3, x4, x5, x6, x7) = U6_gaaaag(x1, x5, x6, x7)
transpose_auxE_in_gga(x1, x2, x3) = transpose_auxE_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x6)
pB_out_gaaaag(x1, x2, x3, x4, x5, x6) = pB_out_gaaaag(x1, x5, x6)
transpose_auxE_out_gga(x1, x2, x3) = transpose_auxE_out_gga(x1, x2)
transposeA_out_ga(x1, x2) = transposeA_out_ga(x1)
TRANSPOSEA_IN_GA(x1, x2) = TRANSPOSEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6) = PB_IN_GAAAAG(x1, x6)
U5_GAAAAG(x1, x2, x3, x4, x5, x6, x7) = U5_GAAAAG(x1, x6, x7)
ROW2COLD_IN_GAAAA(x1, x2, x3, x4, x5) = ROW2COLD_IN_GAAAA(x1)
U3_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U3_GAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x21)
ROW2COLC_IN_GAAGA(x1, x2, x3, x4, x5) = ROW2COLC_IN_GAAGA(x1, x4)
U2_GAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_GAAGA(x1, x2, x6, x8)
U6_GAAAAG(x1, x2, x3, x4, x5, x6, x7) = U6_GAAAAG(x1, x5, x6, x7)
TRANSPOSE_AUXE_IN_GGA(x1, x2, x3) = TRANSPOSE_AUXE_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x6)

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

(4) Obligation:

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

TRANSPOSEA_IN_GA(.(T24, T25), .(T28, T29)) → U1_GA(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
TRANSPOSEA_IN_GA(.(T24, T25), .(T28, T29)) → PB_IN_GAAAAG(T24, T28, T29, X35, X36, T25)
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → ROW2COLD_IN_GAAAA(T24, T28, T29, T34, T35)
ROW2COLD_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, X476)))))))), X477) → U3_GAAAA(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_in_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477))
ROW2COLD_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, X476)))))))), X477) → ROW2COLC_IN_GAAGA(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477)
ROW2COLC_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → U2_GAAGA(T266, T267, T268, T271, X539, T270, X540, row2colC_in_gaaga(T267, T271, X539, .([], T270), X540))
ROW2COLC_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → ROW2COLC_IN_GAAGA(T267, T271, X539, .([], T270), X540)
U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_GAAAAG(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35, T34)
TRANSPOSE_AUXE_IN_GGA(.(T305, T306), T307, .(T310, T311)) → U4_GGA(T305, T306, T307, T310, T311, pB_in_gaaaag(T305, T310, T311, X611, X612, T306))
TRANSPOSE_AUXE_IN_GGA(.(T305, T306), T307, .(T310, T311)) → PB_IN_GAAAAG(T305, T310, T311, X611, X612, T306)

The TRS R consists of the following rules:

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_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, X476)))))))), X477) → U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_in_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477))
row2colC_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_in_gaaga(T267, T271, X539, .([], T270), X540))
row2colC_in_gaaga([], [], [], T279, .([], T279)) → row2colC_out_gaaga([], [], [], T279, .([], T279))
U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_out_gaaga(T267, T271, X539, .([], T270), X540)) → row2colC_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540)
U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_out_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477)) → row2colD_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, X476)))))))), X477)
row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2colD_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2colD_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2colD_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2colD_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T305, T306), T307, .(T310, T311)) → U4_gga(T305, T306, T307, T310, T311, pB_in_gaaaag(T305, T310, T311, X611, X612, T306))
U4_gga(T305, T306, T307, T310, T311, pB_out_gaaaag(T305, T310, T311, X611, X612, T306)) → transpose_auxE_out_gga(.(T305, T306), T307, .(T310, T311))
transpose_auxE_in_gga([], T318, T318) → transpose_auxE_out_gga([], T318, T318)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ROW2COLC_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → ROW2COLC_IN_GAAGA(T267, T271, X539, .([], T270), X540)

The TRS R consists of the following rules:

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_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, X476)))))))), X477) → U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_in_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477))
row2colC_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_in_gaaga(T267, T271, X539, .([], T270), X540))
row2colC_in_gaaga([], [], [], T279, .([], T279)) → row2colC_out_gaaga([], [], [], T279, .([], T279))
U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_out_gaaga(T267, T271, X539, .([], T270), X540)) → row2colC_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540)
U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_out_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477)) → row2colD_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, X476)))))))), X477)
row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2colD_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2colD_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2colD_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2colD_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T305, T306), T307, .(T310, T311)) → U4_gga(T305, T306, T307, T310, T311, pB_in_gaaaag(T305, T310, T311, X611, X612, T306))
U4_gga(T305, T306, T307, T310, T311, pB_out_gaaaag(T305, T310, T311, X611, X612, T306)) → transpose_auxE_out_gga(.(T305, T306), T307, .(T310, T311))
transpose_auxE_in_gga([], T318, T318) → transpose_auxE_out_gga([], T318, T318)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

(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:

ROW2COLC_IN_GAAGA(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → ROW2COLC_IN_GAAGA(T267, T271, X539, .([], T270), X540)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COLC_IN_GAAGA(x1, x2, x3, x4, x5) = ROW2COLC_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:

ROW2COLC_IN_GAAGA(.(T266, T267), T270) → ROW2COLC_IN_GAAGA(T267, .([], T270))

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:

ROW2COLC_IN_GAAGA(.(T266, T267), T270) → ROW2COLC_IN_GAAGA(T267, .([], T270))
The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35, T34)
TRANSPOSE_AUXE_IN_GGA(.(T305, T306), T307, .(T310, T311)) → PB_IN_GAAAAG(T305, T310, T311, X611, X612, T306)
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))

The TRS R consists of the following rules:

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_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, X476)))))))), X477) → U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_in_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477))
row2colC_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_in_gaaga(T267, T271, X539, .([], T270), X540))
row2colC_in_gaaga([], [], [], T279, .([], T279)) → row2colC_out_gaaga([], [], [], T279, .([], T279))
U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_out_gaaga(T267, T271, X539, .([], T270), X540)) → row2colC_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540)
U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_out_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477)) → row2colD_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, X476)))))))), X477)
row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2colD_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2colD_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2colD_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2colD_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T305, T306), T307, .(T310, T311)) → U4_gga(T305, T306, T307, T310, T311, pB_in_gaaaag(T305, T310, T311, X611, X612, T306))
U4_gga(T305, T306, T307, T310, T311, pB_out_gaaaag(T305, T310, T311, X611, X612, T306)) → transpose_auxE_out_gga(.(T305, T306), T307, .(T310, T311))
transpose_auxE_in_gga([], T318, T318) → transpose_auxE_out_gga([], T318, T318)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_ga(x1, x2) = transposeA_in_ga(x1)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6) = pB_in_gaaaag(x1, x6)
U5_gaaaag(x1, x2, x3, x4, x5, x6, x7) = U5_gaaaag(x1, x6, x7)
row2colD_in_gaaaa(x1, x2, x3, x4, x5) = row2colD_in_gaaaa(x1)
U3_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U3_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x21)
row2colC_in_gaaga(x1, x2, x3, x4, x5) = row2colC_in_gaaga(x1, x4)
U2_gaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_gaaga(x1, x2, x6, x8)
[] = []
row2colC_out_gaaga(x1, x2, x3, x4, x5) = row2colC_out_gaaga(x1, x4, x5)
row2colD_out_gaaaa(x1, x2, x3, x4, x5) = row2colD_out_gaaaa(x1, x5)
U6_gaaaag(x1, x2, x3, x4, x5, x6, x7) = U6_gaaaag(x1, x5, x6, x7)
transpose_auxE_in_gga(x1, x2, x3) = transpose_auxE_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x6)
pB_out_gaaaag(x1, x2, x3, x4, x5, x6) = pB_out_gaaaag(x1, x5, x6)
transpose_auxE_out_gga(x1, x2, x3) = transpose_auxE_out_gga(x1, x2)
transposeA_out_ga(x1, x2) = transposeA_out_ga(x1)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6) = PB_IN_GAAAAG(x1, x6)
U5_GAAAAG(x1, x2, x3, x4, x5, x6, x7) = U5_GAAAAG(x1, x6, x7)
TRANSPOSE_AUXE_IN_GGA(x1, x2, x3) = TRANSPOSE_AUXE_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35, T34)
TRANSPOSE_AUXE_IN_GGA(.(T305, T306), T307, .(T310, T311)) → PB_IN_GAAAAG(T305, T310, T311, X611, X612, T306)
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))

The TRS R consists of the following rules:

row2colD_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, X476)))))))), X477) → U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_in_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477))
row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], [])))))))) → row2colD_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, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], []))))))) → row2colD_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, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], [])))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), .(.(T156, T158), [])))), .(T58, .(T83, .(T108, .(T133, .(T158, []))))), .([], .([], .([], .([], .([], []))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], []))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), .(.(T131, T133), []))), .(T58, .(T83, .(T108, .(T133, [])))), .([], .([], .([], .([], [])))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], [])))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, []))), .(T56, T58), .(.(T81, T83), .(.(T106, T108), [])), .(T58, .(T83, .(T108, []))), .([], .([], .([], []))))
row2colD_in_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], []))) → row2colD_out_gaaaa(.(T56, .(T81, [])), .(T56, T58), .(.(T81, T83), []), .(T58, .(T83, [])), .([], .([], [])))
row2colD_in_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], [])) → row2colD_out_gaaaa(.(T56, []), .(T56, T58), [], .(T58, []), .([], []))
U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, T58, T83, T108, T133, T158, T183, T208, T233, T235, X476, X477, row2colC_out_gaaga(T232, T235, X476, .([], .([], .([], .([], .([], .([], .([], []))))))), X477)) → row2colD_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, X476)))))))), X477)
row2colC_in_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540) → U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_in_gaaga(T267, T271, X539, .([], T270), X540))
row2colC_in_gaaga([], [], [], T279, .([], T279)) → row2colC_out_gaaga([], [], [], T279, .([], T279))
U2_gaaga(T266, T267, T268, T271, X539, T270, X540, row2colC_out_gaaga(T267, T271, X539, .([], T270), X540)) → row2colC_out_gaaga(.(T266, T267), .(.(T266, T268), T271), .(T268, X539), T270, X540)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
row2colD_in_gaaaa(x1, x2, x3, x4, x5) = row2colD_in_gaaaa(x1)
U3_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U3_gaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x21)
row2colC_in_gaaga(x1, x2, x3, x4, x5) = row2colC_in_gaaga(x1, x4)
U2_gaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_gaaga(x1, x2, x6, x8)
[] = []
row2colC_out_gaaga(x1, x2, x3, x4, x5) = row2colC_out_gaaga(x1, x4, x5)
row2colD_out_gaaaa(x1, x2, x3, x4, x5) = row2colD_out_gaaaa(x1, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6) = PB_IN_GAAAAG(x1, x6)
U5_GAAAAG(x1, x2, x3, x4, x5, x6, x7) = U5_GAAAAG(x1, x6, x7)
TRANSPOSE_AUXE_IN_GGA(x1, x2, x3) = TRANSPOSE_AUXE_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

U5_GAAAAG(T24, T25, row2colD_out_gaaaa(T24, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35)
TRANSPOSE_AUXE_IN_GGA(.(T305, T306), T307) → PB_IN_GAAAAG(T305, T306)
PB_IN_GAAAAG(T24, T25) → U5_GAAAAG(T24, T25, row2colD_in_gaaaa(T24))

The TRS R consists of the following rules:

row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232))))))))) → U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, row2colC_in_gaaga(T232, .([], .([], .([], .([], .([], .([], .([], [])))))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, [])))))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, []))))))), .([], .([], .([], .([], .([], .([], .([], []))))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, []))))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, [])))))), .([], .([], .([], .([], .([], .([], [])))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, [])))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, []))))), .([], .([], .([], .([], .([], []))))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, .(T131, []))))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, [])))), .([], .([], .([], .([], [])))))
row2colD_in_gaaaa(.(T56, .(T81, .(T106, [])))) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, []))), .([], .([], .([], []))))
row2colD_in_gaaaa(.(T56, .(T81, []))) → row2colD_out_gaaaa(.(T56, .(T81, [])), .([], .([], [])))
row2colD_in_gaaaa(.(T56, [])) → row2colD_out_gaaaa(.(T56, []), .([], []))
U3_gaaaa(T56, T81, T106, T131, T156, T181, T206, T231, T232, row2colC_out_gaaga(T232, .([], .([], .([], .([], .([], .([], .([], []))))))), X477)) → row2colD_out_gaaaa(.(T56, .(T81, .(T106, .(T131, .(T156, .(T181, .(T206, .(T231, T232)))))))), X477)
row2colC_in_gaaga(.(T266, T267), T270) → U2_gaaga(T266, T267, T270, row2colC_in_gaaga(T267, .([], T270)))
row2colC_in_gaaga([], T279) → row2colC_out_gaaga([], T279, .([], T279))
U2_gaaga(T266, T267, T270, row2colC_out_gaaga(T267, .([], T270), X540)) → row2colC_out_gaaga(.(T266, T267), T270, X540)

The set Q consists of the following terms:

row2colD_in_gaaaa(x0)
U3_gaaaa(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)
row2colC_in_gaaga(x0, x1)
U2_gaaga(x0, x1, x2, x3)

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

(19) 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:

TRANSPOSE_AUXE_IN_GGA(.(T305, T306), T307) → PB_IN_GAAAAG(T305, T306)
The graph contains the following edges 1 > 1, 1 > 2
PB_IN_GAAAAG(T24, T25) → U5_GAAAAG(T24, T25, row2colD_in_gaaaa(T24))
The graph contains the following edges 1 >= 1, 2 >= 2
U5_GAAAAG(T24, T25, row2colD_out_gaaaa(T24, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35)
The graph contains the following edges 2 >= 1, 3 > 2

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT 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) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES