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

(0) Obligation:

Clauses:

f(A, [], RES) :- g(A, [], RES).
f(.(A, As), .(B, Bs), RES) :- f(.(B, .(A, As)), Bs, RES).
g([], RES, RES).
g(.(C, Cs), D, RES) :- g(Cs, .(C, D), RES).

Query: f(g,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: f(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: f(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | f(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: g(T6, [], T8) | f(T6, [], T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: f(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\nf(T1, T2, T3) !=? f(X3, [], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: g(T6, [], T8), SCOPE: 2, CLAUSE: 2 | g(T6, [], T8), SCOPE: 2, CLAUSE: 3 | ?_2 | f(T6, [], T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: g(T6, [], T8), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: g(T6, [], T8), SCOPE: 2, CLAUSE: 3 | ?_2 | f(T6, [], T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: g(T6, [], T8), SCOPE: 2, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ?_2 | f(T6, [], T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="B: g(T22, .(T21, []), T24)\n\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: g(T22, .(T21, []), T24), SCOPE: 3, CLAUSE: 2 | g(T22, .(T21, []), T24), SCOPE: 3, CLAUSE: 3\n\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: g(T22, .(T21, []), T24), SCOPE: 3, CLAUSE: 2\n\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: g(T22, .(T21, []), T24), SCOPE: 3, CLAUSE: 3\n\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: g(T41, .(T40, .(T42, [])), T44)\n\nT40 is ground\nT41 is ground\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: g(T41, .(T40, .(T42, [])), T44), SCOPE: 4, CLAUSE: 2 | g(T41, .(T40, .(T42, [])), T44), SCOPE: 4, CLAUSE: 3\n\nT40 is ground\nT41 is ground\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: g(T41, .(T40, .(T42, [])), T44), SCOPE: 4, CLAUSE: 2\n\nT40 is ground\nT41 is ground\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: g(T41, .(T40, .(T42, [])), T44), SCOPE: 4, CLAUSE: 3\n\nT40 is ground\nT41 is ground\nT42 is ground", 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];
74 [label="74: g(T70, .(T69, .(T71, .(T72, []))), T74)\n\nT69 is ground\nT70 is ground\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: g(T70, .(T69, .(T71, .(T72, []))), T74), SCOPE: 5, CLAUSE: 2 | g(T70, .(T69, .(T71, .(T72, []))), T74), SCOPE: 5, CLAUSE: 3\n\nT69 is ground\nT70 is ground\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: g(T70, .(T69, .(T71, .(T72, []))), T74), SCOPE: 5, CLAUSE: 2\n\nT69 is ground\nT70 is ground\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: g(T70, .(T69, .(T71, .(T72, []))), T74), SCOPE: 5, CLAUSE: 3\n\nT69 is ground\nT70 is ground\nT71 is ground\nT72 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: g(T109, .(T108, .(T110, .(T111, .(T112, [])))), T114)\n\nT108 is ground\nT109 is ground\nT110 is ground\nT111 is ground\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: g(T109, .(T108, .(T110, .(T111, .(T112, [])))), T114), SCOPE: 6, CLAUSE: 2 | g(T109, .(T108, .(T110, .(T111, .(T112, [])))), T114), SCOPE: 6, CLAUSE: 3\n\nT108 is ground\nT109 is ground\nT110 is ground\nT111 is ground\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: g(T109, .(T108, .(T110, .(T111, .(T112, [])))), T114), SCOPE: 6, CLAUSE: 2\n\nT108 is ground\nT109 is ground\nT110 is ground\nT111 is ground\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: g(T109, .(T108, .(T110, .(T111, .(T112, [])))), T114), SCOPE: 6, CLAUSE: 3\n\nT108 is ground\nT109 is ground\nT110 is ground\nT111 is ground\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: g(T158, .(T157, .(T159, .(T160, .(T161, .(T162, []))))), T164)\n\nT157 is ground\nT158 is ground\nT159 is ground\nT160 is ground\nT161 is ground\nT162 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: g(T158, .(T157, .(T159, .(T160, .(T161, .(T162, []))))), T164), SCOPE: 7, CLAUSE: 2 | g(T158, .(T157, .(T159, .(T160, .(T161, .(T162, []))))), T164), SCOPE: 7, CLAUSE: 3\n\nT157 is ground\nT158 is ground\nT159 is ground\nT160 is ground\nT161 is ground\nT162 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: g(T158, .(T157, .(T159, .(T160, .(T161, .(T162, []))))), T164), SCOPE: 7, CLAUSE: 2\n\nT157 is ground\nT158 is ground\nT159 is ground\nT160 is ground\nT161 is ground\nT162 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: g(T158, .(T157, .(T159, .(T160, .(T161, .(T162, []))))), T164), SCOPE: 7, CLAUSE: 3\n\nT157 is ground\nT158 is ground\nT159 is ground\nT160 is ground\nT161 is ground\nT162 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: g(T217, .(T216, .(T218, .(T219, .(T220, .(T221, .(T222, [])))))), T224)\n\nT216 is ground\nT217 is ground\nT218 is ground\nT219 is ground\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
150 [label="150: g(T217, .(T216, .(T218, .(T219, .(T220, .(T221, .(T222, [])))))), T224), SCOPE: 8, CLAUSE: 2 | g(T217, .(T216, .(T218, .(T219, .(T220, .(T221, .(T222, [])))))), T224), SCOPE: 8, CLAUSE: 3\n\nT216 is ground\nT217 is ground\nT218 is ground\nT219 is ground\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: g(T217, .(T216, .(T218, .(T219, .(T220, .(T221, .(T222, [])))))), T224), SCOPE: 8, CLAUSE: 2\n\nT216 is ground\nT217 is ground\nT218 is ground\nT219 is ground\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: g(T217, .(T216, .(T218, .(T219, .(T220, .(T221, .(T222, [])))))), T224), SCOPE: 8, CLAUSE: 3\n\nT216 is ground\nT217 is ground\nT218 is ground\nT219 is ground\nT220 is ground\nT221 is ground\nT222 is ground", 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];
164 [label="164: g(T286, .(T285, .(T287, .(T288, .(T289, .(T290, .(T291, .(T292, []))))))), T294)\n\nT285 is ground\nT286 is ground\nT287 is ground\nT288 is ground\nT289 is ground\nT290 is ground\nT291 is ground\nT292 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
165 [label="165: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: g(T286, .(T285, .(T287, .(T288, .(T289, .(T290, .(T291, .(T292, []))))))), T294), SCOPE: 9, CLAUSE: 2 | g(T286, .(T285, .(T287, .(T288, .(T289, .(T290, .(T291, .(T292, []))))))), T294), SCOPE: 9, CLAUSE: 3\n\nT285 is ground\nT286 is ground\nT287 is ground\nT288 is ground\nT289 is ground\nT290 is ground\nT291 is ground\nT292 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: g(T286, .(T285, .(T287, .(T288, .(T289, .(T290, .(T291, .(T292, []))))))), T294), SCOPE: 9, CLAUSE: 2\n\nT285 is ground\nT286 is ground\nT287 is ground\nT288 is ground\nT289 is ground\nT290 is ground\nT291 is ground\nT292 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: g(T286, .(T285, .(T287, .(T288, .(T289, .(T290, .(T291, .(T292, []))))))), T294), SCOPE: 9, CLAUSE: 3\n\nT285 is ground\nT286 is ground\nT287 is ground\nT288 is ground\nT289 is ground\nT290 is ground\nT291 is ground\nT292 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
178 [label="178: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
180 [label="180: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
189 [label="189: g(T365, .(T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, [])))))))), T374)\n\nT364 is ground\nT365 is ground\nT366 is ground\nT367 is ground\nT368 is ground\nT369 is ground\nT370 is ground\nT371 is ground\nT372 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
190 [label="190: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="C: g(T365, .(T364, T391), T374)\n\nT364 is ground\nT365 is ground\nT391 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
194 [label="194: g(T365, .(T364, T391), T374), SCOPE: 10, CLAUSE: 2 | g(T365, .(T364, T391), T374), SCOPE: 10, CLAUSE: 3\n\nT364 is ground\nT365 is ground\nT391 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="197: g(T365, .(T364, T391), T374), SCOPE: 10, CLAUSE: 2\n\nT364 is ground\nT365 is ground\nT391 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="198: g(T365, .(T364, T391), T374), SCOPE: 10, CLAUSE: 3\n\nT364 is ground\nT365 is ground\nT391 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="201: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
207 [label="207: g(T417, .(T416, .(T418, T419)), T421)\n\nT416 is ground\nT417 is ground\nT418 is ground\nT419 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
208 [label="208: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="212: f(T6, [], T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
220 [label="220: f(.(T437, .(T435, T436)), T438, T440)\n\nT435 is ground\nT436 is ground\nT437 is ground\nT438 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: f(.(T437, .(T435, T436)), T438, T440), SCOPE: 11, CLAUSE: 0 | f(.(T437, .(T435, T436)), T438, T440), SCOPE: 11, CLAUSE: 1\n\nT435 is ground\nT436 is ground\nT437 is ground\nT438 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: f(.(T437, .(T435, T436)), T438, T440), SCOPE: 11, CLAUSE: 0\n\nT435 is ground\nT436 is ground\nT437 is ground\nT438 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
224 [label="224: f(.(T437, .(T435, T436)), T438, T440), SCOPE: 11, CLAUSE: 1\n\nT435 is ground\nT436 is ground\nT437 is ground\nT438 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="231: g(.(T457, .(T458, T459)), [], T461)\n\nT457 is ground\nT458 is ground\nT459 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
235 [label="235: g(.(T457, .(T458, T459)), [], T461), SCOPE: 12, CLAUSE: 2 | g(.(T457, .(T458, T459)), [], T461), SCOPE: 12, CLAUSE: 3\n\nT457 is ground\nT458 is ground\nT459 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
236 [label="236: g(.(T457, .(T458, T459)), [], T461), SCOPE: 12, CLAUSE: 3\n\nT457 is ground\nT458 is ground\nT459 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
241 [label="241: g(.(T472, T473), .(T471, []), T475)\n\nT471 is ground\nT472 is ground\nT473 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
245 [label="245: f(.(T491, .(T488, .(T489, T490))), T492, T494)\n\nT488 is ground\nT489 is ground\nT490 is ground\nT491 is ground\nT492 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
246 [label="246: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
247 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 247 [arrowhead = none ];
247 -> 5;

248 [label="EVAL with clause\nf(X3, [], X4) :- g(X3, [], X4).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> [],\nT3 -> T8,\nX4 -> T8,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 248 [arrowhead = none ];
248 -> 10;

249 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 249 [arrowhead = none ];
249 -> 11;

250 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 250 [arrowhead = none ];
250 -> 15;

251 [label="EVAL with clause\nf(.(X196, X197), .(X198, X199), X200) :- f(.(X198, .(X196, X197)), X199, X200).\nand substitutionX196 -> T435,\nX197 -> T436,\nT1 -> .(T435, T436),\nX198 -> T437,\nX199 -> T438,\nT2 -> .(T437, T438),\nT3 -> T440,\nX200 -> T440,\nT439 -> T440", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 251 [arrowhead = none ];
251 -> 220;

252 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 252 [arrowhead = none ];
252 -> 221;

253 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 253 [arrowhead = none ];
253 -> 17;

254 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 254 [arrowhead = none ];
254 -> 19;

255 [label="EVAL with clause\ng([], X9, X9).\nand substitutionT6 -> [],\nX9 -> [],\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 255 [arrowhead = none ];
255 -> 21;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 256 [arrowhead = none ];
256 -> 23;

257 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 257 [arrowhead = none ];
257 -> 28;

258 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 258 [arrowhead = none ];
258 -> 31;

259 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 259 [arrowhead = none ];
259 -> 25;

260 [label="EVAL with clause\ng(.(X26, X27), X28, X29) :- g(X27, .(X26, X28), X29).\nand substitutionX26 -> T21,\nX27 -> T22,\nT6 -> .(T21, T22),\nX28 -> [],\nT8 -> T24,\nX29 -> T24,\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 260 [arrowhead = none ];
260 -> 37;

261 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 261 [arrowhead = none ];
261 -> 38;

262 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 262 [arrowhead = none ];
262 -> 212;

263 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 263 [arrowhead = none ];
263 -> 43;

264 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 264 [arrowhead = none ];
264 -> 44;

265 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 265 [arrowhead = none ];
265 -> 45;

266 [label="EVAL with clause\ng([], X36, X36).\nand substitutionT22 -> [],\nT21 -> T31,\nX36 -> .(T31, []),\nT24 -> .(T31, [])", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 266 [arrowhead = none ];
266 -> 47;

267 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 267 [arrowhead = none ];
267 -> 48;

268 [label="EVAL with clause\ng(.(X45, X46), X47, X48) :- g(X46, .(X45, X47), X48).\nand substitutionX45 -> T40,\nX46 -> T41,\nT22 -> .(T40, T41),\nT21 -> T42,\nX47 -> .(T42, []),\nT24 -> T44,\nX48 -> T44,\nT43 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 268 [arrowhead = none ];
268 -> 58;

269 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 269 [arrowhead = none ];
269 -> 59;

270 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 270 [arrowhead = none ];
270 -> 49;

271 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 271 [arrowhead = none ];
271 -> 63;

272 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 272 [arrowhead = none ];
272 -> 64;

273 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 273 [arrowhead = none ];
273 -> 65;

274 [label="EVAL with clause\ng([], X55, X55).\nand substitutionT41 -> [],\nT40 -> T57,\nT42 -> T58,\nX55 -> .(T57, .(T58, [])),\nT44 -> .(T57, .(T58, []))", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 274 [arrowhead = none ];
274 -> 66;

275 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 275 [arrowhead = none ];
275 -> 67;

276 [label="EVAL with clause\ng(.(X64, X65), X66, X67) :- g(X65, .(X64, X66), X67).\nand substitutionX64 -> T69,\nX65 -> T70,\nT41 -> .(T69, T70),\nT40 -> T71,\nT42 -> T72,\nX66 -> .(T71, .(T72, [])),\nT44 -> T74,\nX67 -> T74,\nT73 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 276 [arrowhead = none ];
276 -> 74;

277 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 277 [arrowhead = none ];
277 -> 76;

278 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 278 [arrowhead = none ];
278 -> 68;

279 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 279 [arrowhead = none ];
279 -> 83;

280 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
83 -> 280 [arrowhead = none ];
280 -> 87;

281 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
83 -> 281 [arrowhead = none ];
281 -> 88;

282 [label="EVAL with clause\ng([], X74, X74).\nand substitutionT70 -> [],\nT69 -> T93,\nT71 -> T94,\nT72 -> T95,\nX74 -> .(T93, .(T94, .(T95, []))),\nT74 -> .(T93, .(T94, .(T95, [])))", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 282 [arrowhead = none ];
282 -> 92;

283 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 283 [arrowhead = none ];
283 -> 96;

284 [label="EVAL with clause\ng(.(X83, X84), X85, X86) :- g(X84, .(X83, X85), X86).\nand substitutionX83 -> T108,\nX84 -> T109,\nT70 -> .(T108, T109),\nT69 -> T110,\nT71 -> T111,\nT72 -> T112,\nX85 -> .(T110, .(T111, .(T112, []))),\nT74 -> T114,\nX86 -> T114,\nT113 -> T114", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 284 [arrowhead = none ];
284 -> 102;

285 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 285 [arrowhead = none ];
285 -> 104;

286 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 286 [arrowhead = none ];
286 -> 97;

287 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
102 -> 287 [arrowhead = none ];
287 -> 106;

288 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 288 [arrowhead = none ];
288 -> 109;

289 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 289 [arrowhead = none ];
289 -> 110;

290 [label="EVAL with clause\ng([], X93, X93).\nand substitutionT109 -> [],\nT108 -> T139,\nT110 -> T140,\nT111 -> T141,\nT112 -> T142,\nX93 -> .(T139, .(T140, .(T141, .(T142, [])))),\nT114 -> .(T139, .(T140, .(T141, .(T142, []))))", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 290 [arrowhead = none ];
290 -> 113;

291 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 291 [arrowhead = none ];
291 -> 115;

292 [label="EVAL with clause\ng(.(X102, X103), X104, X105) :- g(X103, .(X102, X104), X105).\nand substitutionX102 -> T157,\nX103 -> T158,\nT109 -> .(T157, T158),\nT108 -> T159,\nT110 -> T160,\nT111 -> T161,\nT112 -> T162,\nX104 -> .(T159, .(T160, .(T161, .(T162, [])))),\nT114 -> T164,\nX105 -> T164,\nT163 -> T164", fontsize=14, style = filled, fillcolor = lightblue];
110 -> 292 [arrowhead = none ];
292 -> 123;

293 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
110 -> 293 [arrowhead = none ];
293 -> 125;

294 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 294 [arrowhead = none ];
294 -> 116;

295 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
123 -> 295 [arrowhead = none ];
295 -> 127;

296 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
127 -> 296 [arrowhead = none ];
296 -> 130;

297 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
127 -> 297 [arrowhead = none ];
297 -> 131;

298 [label="EVAL with clause\ng([], X112, X112).\nand substitutionT158 -> [],\nT157 -> T195,\nT159 -> T196,\nT160 -> T197,\nT161 -> T198,\nT162 -> T199,\nX112 -> .(T195, .(T196, .(T197, .(T198, .(T199, []))))),\nT164 -> .(T195, .(T196, .(T197, .(T198, .(T199, [])))))", fontsize=14, style = filled, fillcolor = lightblue];
130 -> 298 [arrowhead = none ];
298 -> 136;

299 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
130 -> 299 [arrowhead = none ];
299 -> 139;

300 [label="EVAL with clause\ng(.(X121, X122), X123, X124) :- g(X122, .(X121, X123), X124).\nand substitutionX121 -> T216,\nX122 -> T217,\nT158 -> .(T216, T217),\nT157 -> T218,\nT159 -> T219,\nT160 -> T220,\nT161 -> T221,\nT162 -> T222,\nX123 -> .(T218, .(T219, .(T220, .(T221, .(T222, []))))),\nT164 -> T224,\nX124 -> T224,\nT223 -> T224", fontsize=14, style = filled, fillcolor = lightblue];
131 -> 300 [arrowhead = none ];
300 -> 143;

301 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
131 -> 301 [arrowhead = none ];
301 -> 146;

302 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
136 -> 302 [arrowhead = none ];
302 -> 140;

303 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
143 -> 303 [arrowhead = none ];
303 -> 150;

304 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
150 -> 304 [arrowhead = none ];
304 -> 153;

305 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
150 -> 305 [arrowhead = none ];
305 -> 155;

306 [label="EVAL with clause\ng([], X131, X131).\nand substitutionT217 -> [],\nT216 -> T261,\nT218 -> T262,\nT219 -> T263,\nT220 -> T264,\nT221 -> T265,\nT222 -> T266,\nX131 -> .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, [])))))),\nT224 -> .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, []))))))", fontsize=14, style = filled, fillcolor = lightblue];
153 -> 306 [arrowhead = none ];
306 -> 156;

307 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
153 -> 307 [arrowhead = none ];
307 -> 157;

308 [label="EVAL with clause\ng(.(X140, X141), X142, X143) :- g(X141, .(X140, X142), X143).\nand substitutionX140 -> T285,\nX141 -> T286,\nT217 -> .(T285, T286),\nT216 -> T287,\nT218 -> T288,\nT219 -> T289,\nT220 -> T290,\nT221 -> T291,\nT222 -> T292,\nX142 -> .(T287, .(T288, .(T289, .(T290, .(T291, .(T292, [])))))),\nT224 -> T294,\nX143 -> T294,\nT293 -> T294", fontsize=14, style = filled, fillcolor = lightblue];
155 -> 308 [arrowhead = none ];
308 -> 164;

309 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
155 -> 309 [arrowhead = none ];
309 -> 165;

310 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
156 -> 310 [arrowhead = none ];
310 -> 158;

311 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
164 -> 311 [arrowhead = none ];
311 -> 169;

312 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
169 -> 312 [arrowhead = none ];
312 -> 172;

313 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
169 -> 313 [arrowhead = none ];
313 -> 173;

314 [label="EVAL with clause\ng([], X150, X150).\nand substitutionT286 -> [],\nT285 -> T337,\nT287 -> T338,\nT288 -> T339,\nT289 -> T340,\nT290 -> T341,\nT291 -> T342,\nT292 -> T343,\nX150 -> .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, []))))))),\nT294 -> .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
172 -> 314 [arrowhead = none ];
314 -> 176;

315 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
172 -> 315 [arrowhead = none ];
315 -> 178;

316 [label="EVAL with clause\ng(.(X159, X160), X161, X162) :- g(X160, .(X159, X161), X162).\nand substitutionX159 -> T364,\nX160 -> T365,\nT286 -> .(T364, T365),\nT285 -> T366,\nT287 -> T367,\nT288 -> T368,\nT289 -> T369,\nT290 -> T370,\nT291 -> T371,\nT292 -> T372,\nX161 -> .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))),\nT294 -> T374,\nX162 -> T374,\nT373 -> T374", fontsize=14, style = filled, fillcolor = lightblue];
173 -> 316 [arrowhead = none ];
316 -> 189;

317 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
173 -> 317 [arrowhead = none ];
317 -> 190;

318 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
176 -> 318 [arrowhead = none ];
318 -> 180;

319 [label="GENERALIZATION\nT391 <-- .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, [])))))))\n\nNew Knowledge:\nT391 is ground", fontsize=14, style = filled, fillcolor = violet];
189 -> 319 [arrowhead = none ];
319 -> 191;

320 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
191 -> 320 [arrowhead = none ];
320 -> 194;

321 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
194 -> 321 [arrowhead = none ];
321 -> 197;

322 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
194 -> 322 [arrowhead = none ];
322 -> 198;

323 [label="EVAL with clause\ng([], X171, X171).\nand substitutionT365 -> [],\nT364 -> T404,\nT391 -> T405,\nX171 -> .(T404, T405),\nT374 -> .(T404, T405)", fontsize=14, style = filled, fillcolor = lightblue];
197 -> 323 [arrowhead = none ];
323 -> 200;

324 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
197 -> 324 [arrowhead = none ];
324 -> 201;

325 [label="EVAL with clause\ng(.(X180, X181), X182, X183) :- g(X181, .(X180, X182), X183).\nand substitutionX180 -> T416,\nX181 -> T417,\nT365 -> .(T416, T417),\nT364 -> T418,\nT391 -> T419,\nX182 -> .(T418, T419),\nT374 -> T421,\nX183 -> T421,\nT420 -> T421", fontsize=14, style = filled, fillcolor = lightblue];
198 -> 325 [arrowhead = none ];
325 -> 207;

326 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
198 -> 326 [arrowhead = none ];
326 -> 208;

327 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
200 -> 327 [arrowhead = none ];
327 -> 202;

328 [label="INSTANCE with matching:\nT365 -> T417\nT364 -> T416\nT391 -> .(T418, T419)\nT374 -> T421", fontsize=14, style = filled, fillcolor = lightgrey];
207 -> 328 [arrowhead = none , style=dashed];
328 -> 191[style=dashed];

329 [label="BACKTRACK\nfor clause: f(.(A, As), .(B, Bs), RES) :- f(.(B, .(A, As)), Bs, RES)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
212 -> 329 [arrowhead = none ];
329 -> 215;

330 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
220 -> 330 [arrowhead = none ];
330 -> 222;

331 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
222 -> 331 [arrowhead = none ];
331 -> 223;

332 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
222 -> 332 [arrowhead = none ];
332 -> 224;

333 [label="EVAL with clause\nf(X209, [], X210) :- g(X209, [], X210).\nand substitutionT437 -> T457,\nT435 -> T458,\nT436 -> T459,\nX209 -> .(T457, .(T458, T459)),\nT438 -> [],\nT440 -> T461,\nX210 -> T461,\nT460 -> T461", fontsize=14, style = filled, fillcolor = lightblue];
223 -> 333 [arrowhead = none ];
333 -> 231;

334 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
223 -> 334 [arrowhead = none ];
334 -> 232;

335 [label="EVAL with clause\nf(.(X240, X241), .(X242, X243), X244) :- f(.(X242, .(X240, X241)), X243, X244).\nand substitutionT437 -> T488,\nX240 -> T488,\nT435 -> T489,\nT436 -> T490,\nX241 -> .(T489, T490),\nX242 -> T491,\nX243 -> T492,\nT438 -> .(T491, T492),\nT440 -> T494,\nX244 -> T494,\nT493 -> T494", fontsize=14, style = filled, fillcolor = lightblue];
224 -> 335 [arrowhead = none ];
335 -> 245;

336 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
224 -> 336 [arrowhead = none ];
336 -> 246;

337 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
231 -> 337 [arrowhead = none ];
337 -> 235;

338 [label="BACKTRACK\nfor clause: g([], RES, RES)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
235 -> 338 [arrowhead = none ];
338 -> 236;

339 [label="ONLY EVAL with clause\ng(.(X224, X225), X226, X227) :- g(X225, .(X224, X226), X227).\nand substitutionT457 -> T471,\nX224 -> T471,\nT458 -> T472,\nT459 -> T473,\nX225 -> .(T472, T473),\nX226 -> [],\nT461 -> T475,\nX227 -> T475,\nT474 -> T475", fontsize=14, style = filled, fillcolor = lightblue];
236 -> 339 [arrowhead = none ];
339 -> 241;

340 [label="INSTANCE with matching:\nT22 -> .(T472, T473)\nT21 -> T471\nT24 -> T475", fontsize=14, style = filled, fillcolor = lightgrey];
241 -> 340 [arrowhead = none , style=dashed];
340 -> 37[style=dashed];

341 [label="INSTANCE with matching:\nT1 -> .(T491, .(T488, .(T489, T490)))\nT2 -> T492\nT3 -> T494", fontsize=14, style = filled, fillcolor = lightgrey];
245 -> 341 [arrowhead = none , style=dashed];
341 -> 1[style=dashed];

}

(2) Obligation:

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

fA_in_gga([], [], []) → fA_out_gga([], [], [])
fA_in_gga(.(T21, T22), [], T24) → U1_gga(T21, T22, T24, gB_in_gga(T22, T21, T24))
gB_in_gga([], T31, .(T31, [])) → gB_out_gga([], T31, .(T31, []))
gB_in_gga(.(T57, []), T58, .(T57, .(T58, []))) → gB_out_gga(.(T57, []), T58, .(T57, .(T58, [])))
gB_in_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, [])))) → gB_out_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, []))))
gB_in_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, []))))) → gB_out_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, [])))))
gB_in_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, [])))))) → gB_out_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, []))))))
gB_in_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, []))))))) → gB_out_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, [])))))))
gB_in_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, [])))))))) → gB_out_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, []))))))))
gB_in_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374) → U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_in_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374))
gC_in_ggga([], T404, T405, .(T404, T405)) → gC_out_ggga([], T404, T405, .(T404, T405))
gC_in_ggga(.(T416, T417), T418, T419, T421) → U4_ggga(T416, T417, T418, T419, T421, gC_in_ggga(T417, T416, .(T418, T419), T421))
U4_ggga(T416, T417, T418, T419, T421, gC_out_ggga(T417, T416, .(T418, T419), T421)) → gC_out_ggga(.(T416, T417), T418, T419, T421)
U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_out_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374)) → gB_out_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374)
U1_gga(T21, T22, T24, gB_out_gga(T22, T21, T24)) → fA_out_gga(.(T21, T22), [], T24)
fA_in_gga(.(T472, T473), .(T471, []), T475) → U2_gga(T472, T473, T471, T475, gB_in_gga(.(T472, T473), T471, T475))
U2_gga(T472, T473, T471, T475, gB_out_gga(.(T472, T473), T471, T475)) → fA_out_gga(.(T472, T473), .(T471, []), T475)
fA_in_gga(.(T489, T490), .(T488, .(T491, T492)), T494) → U3_gga(T489, T490, T488, T491, T492, T494, fA_in_gga(.(T491, .(T488, .(T489, T490))), T492, T494))
U3_gga(T489, T490, T488, T491, T492, T494, fA_out_gga(.(T491, .(T488, .(T489, T490))), T492, T494)) → fA_out_gga(.(T489, T490), .(T488, .(T491, T492)), T494)

The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3)  =  fA_in_gga(x1, x2)
[]  =  []
fA_out_gga(x1, x2, x3)  =  fA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
gB_in_gga(x1, x2, x3)  =  gB_in_gga(x1, x2)
gB_out_gga(x1, x2, x3)  =  gB_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)  =  U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
gC_in_ggga(x1, x2, x3, x4)  =  gC_in_ggga(x1, x2, x3)
gC_out_ggga(x1, x2, x3, x4)  =  gC_out_ggga(x1, x2, x3, x4)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gga(x1, x2, x3, x4, x5, x7)

(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:

FA_IN_GGA(.(T21, T22), [], T24) → U1_GGA(T21, T22, T24, gB_in_gga(T22, T21, T24))
FA_IN_GGA(.(T21, T22), [], T24) → GB_IN_GGA(T22, T21, T24)
GB_IN_GGA(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374) → U5_GGA(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_in_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374))
GB_IN_GGA(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374) → GC_IN_GGGA(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374)
GC_IN_GGGA(.(T416, T417), T418, T419, T421) → U4_GGGA(T416, T417, T418, T419, T421, gC_in_ggga(T417, T416, .(T418, T419), T421))
GC_IN_GGGA(.(T416, T417), T418, T419, T421) → GC_IN_GGGA(T417, T416, .(T418, T419), T421)
FA_IN_GGA(.(T472, T473), .(T471, []), T475) → U2_GGA(T472, T473, T471, T475, gB_in_gga(.(T472, T473), T471, T475))
FA_IN_GGA(.(T472, T473), .(T471, []), T475) → GB_IN_GGA(.(T472, T473), T471, T475)
FA_IN_GGA(.(T489, T490), .(T488, .(T491, T492)), T494) → U3_GGA(T489, T490, T488, T491, T492, T494, fA_in_gga(.(T491, .(T488, .(T489, T490))), T492, T494))
FA_IN_GGA(.(T489, T490), .(T488, .(T491, T492)), T494) → FA_IN_GGA(.(T491, .(T488, .(T489, T490))), T492, T494)

The TRS R consists of the following rules:

fA_in_gga([], [], []) → fA_out_gga([], [], [])
fA_in_gga(.(T21, T22), [], T24) → U1_gga(T21, T22, T24, gB_in_gga(T22, T21, T24))
gB_in_gga([], T31, .(T31, [])) → gB_out_gga([], T31, .(T31, []))
gB_in_gga(.(T57, []), T58, .(T57, .(T58, []))) → gB_out_gga(.(T57, []), T58, .(T57, .(T58, [])))
gB_in_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, [])))) → gB_out_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, []))))
gB_in_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, []))))) → gB_out_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, [])))))
gB_in_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, [])))))) → gB_out_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, []))))))
gB_in_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, []))))))) → gB_out_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, [])))))))
gB_in_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, [])))))))) → gB_out_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, []))))))))
gB_in_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374) → U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_in_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374))
gC_in_ggga([], T404, T405, .(T404, T405)) → gC_out_ggga([], T404, T405, .(T404, T405))
gC_in_ggga(.(T416, T417), T418, T419, T421) → U4_ggga(T416, T417, T418, T419, T421, gC_in_ggga(T417, T416, .(T418, T419), T421))
U4_ggga(T416, T417, T418, T419, T421, gC_out_ggga(T417, T416, .(T418, T419), T421)) → gC_out_ggga(.(T416, T417), T418, T419, T421)
U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_out_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374)) → gB_out_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374)
U1_gga(T21, T22, T24, gB_out_gga(T22, T21, T24)) → fA_out_gga(.(T21, T22), [], T24)
fA_in_gga(.(T472, T473), .(T471, []), T475) → U2_gga(T472, T473, T471, T475, gB_in_gga(.(T472, T473), T471, T475))
U2_gga(T472, T473, T471, T475, gB_out_gga(.(T472, T473), T471, T475)) → fA_out_gga(.(T472, T473), .(T471, []), T475)
fA_in_gga(.(T489, T490), .(T488, .(T491, T492)), T494) → U3_gga(T489, T490, T488, T491, T492, T494, fA_in_gga(.(T491, .(T488, .(T489, T490))), T492, T494))
U3_gga(T489, T490, T488, T491, T492, T494, fA_out_gga(.(T491, .(T488, .(T489, T490))), T492, T494)) → fA_out_gga(.(T489, T490), .(T488, .(T491, T492)), T494)

The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3)  =  fA_in_gga(x1, x2)
[]  =  []
fA_out_gga(x1, x2, x3)  =  fA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
gB_in_gga(x1, x2, x3)  =  gB_in_gga(x1, x2)
gB_out_gga(x1, x2, x3)  =  gB_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)  =  U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
gC_in_ggga(x1, x2, x3, x4)  =  gC_in_ggga(x1, x2, x3)
gC_out_ggga(x1, x2, x3, x4)  =  gC_out_ggga(x1, x2, x3, x4)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gga(x1, x2, x3, x4, x5, x7)
FA_IN_GGA(x1, x2, x3)  =  FA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
GB_IN_GGA(x1, x2, x3)  =  GB_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)  =  U5_GGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
GC_IN_GGGA(x1, x2, x3, x4)  =  GC_IN_GGGA(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U3_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GGA(x1, x2, x3, x4, x5, x7)

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

(4) Obligation:

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

FA_IN_GGA(.(T21, T22), [], T24) → U1_GGA(T21, T22, T24, gB_in_gga(T22, T21, T24))
FA_IN_GGA(.(T21, T22), [], T24) → GB_IN_GGA(T22, T21, T24)
GB_IN_GGA(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374) → U5_GGA(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_in_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374))
GB_IN_GGA(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374) → GC_IN_GGGA(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374)
GC_IN_GGGA(.(T416, T417), T418, T419, T421) → U4_GGGA(T416, T417, T418, T419, T421, gC_in_ggga(T417, T416, .(T418, T419), T421))
GC_IN_GGGA(.(T416, T417), T418, T419, T421) → GC_IN_GGGA(T417, T416, .(T418, T419), T421)
FA_IN_GGA(.(T472, T473), .(T471, []), T475) → U2_GGA(T472, T473, T471, T475, gB_in_gga(.(T472, T473), T471, T475))
FA_IN_GGA(.(T472, T473), .(T471, []), T475) → GB_IN_GGA(.(T472, T473), T471, T475)
FA_IN_GGA(.(T489, T490), .(T488, .(T491, T492)), T494) → U3_GGA(T489, T490, T488, T491, T492, T494, fA_in_gga(.(T491, .(T488, .(T489, T490))), T492, T494))
FA_IN_GGA(.(T489, T490), .(T488, .(T491, T492)), T494) → FA_IN_GGA(.(T491, .(T488, .(T489, T490))), T492, T494)

The TRS R consists of the following rules:

fA_in_gga([], [], []) → fA_out_gga([], [], [])
fA_in_gga(.(T21, T22), [], T24) → U1_gga(T21, T22, T24, gB_in_gga(T22, T21, T24))
gB_in_gga([], T31, .(T31, [])) → gB_out_gga([], T31, .(T31, []))
gB_in_gga(.(T57, []), T58, .(T57, .(T58, []))) → gB_out_gga(.(T57, []), T58, .(T57, .(T58, [])))
gB_in_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, [])))) → gB_out_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, []))))
gB_in_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, []))))) → gB_out_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, [])))))
gB_in_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, [])))))) → gB_out_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, []))))))
gB_in_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, []))))))) → gB_out_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, [])))))))
gB_in_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, [])))))))) → gB_out_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, []))))))))
gB_in_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374) → U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_in_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374))
gC_in_ggga([], T404, T405, .(T404, T405)) → gC_out_ggga([], T404, T405, .(T404, T405))
gC_in_ggga(.(T416, T417), T418, T419, T421) → U4_ggga(T416, T417, T418, T419, T421, gC_in_ggga(T417, T416, .(T418, T419), T421))
U4_ggga(T416, T417, T418, T419, T421, gC_out_ggga(T417, T416, .(T418, T419), T421)) → gC_out_ggga(.(T416, T417), T418, T419, T421)
U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_out_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374)) → gB_out_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374)
U1_gga(T21, T22, T24, gB_out_gga(T22, T21, T24)) → fA_out_gga(.(T21, T22), [], T24)
fA_in_gga(.(T472, T473), .(T471, []), T475) → U2_gga(T472, T473, T471, T475, gB_in_gga(.(T472, T473), T471, T475))
U2_gga(T472, T473, T471, T475, gB_out_gga(.(T472, T473), T471, T475)) → fA_out_gga(.(T472, T473), .(T471, []), T475)
fA_in_gga(.(T489, T490), .(T488, .(T491, T492)), T494) → U3_gga(T489, T490, T488, T491, T492, T494, fA_in_gga(.(T491, .(T488, .(T489, T490))), T492, T494))
U3_gga(T489, T490, T488, T491, T492, T494, fA_out_gga(.(T491, .(T488, .(T489, T490))), T492, T494)) → fA_out_gga(.(T489, T490), .(T488, .(T491, T492)), T494)

The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3)  =  fA_in_gga(x1, x2)
[]  =  []
fA_out_gga(x1, x2, x3)  =  fA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
gB_in_gga(x1, x2, x3)  =  gB_in_gga(x1, x2)
gB_out_gga(x1, x2, x3)  =  gB_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)  =  U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
gC_in_ggga(x1, x2, x3, x4)  =  gC_in_ggga(x1, x2, x3)
gC_out_ggga(x1, x2, x3, x4)  =  gC_out_ggga(x1, x2, x3, x4)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gga(x1, x2, x3, x4, x5, x7)
FA_IN_GGA(x1, x2, x3)  =  FA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
GB_IN_GGA(x1, x2, x3)  =  GB_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)  =  U5_GGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
GC_IN_GGGA(x1, x2, x3, x4)  =  GC_IN_GGGA(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U3_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GGA(x1, x2, x3, x4, x5, x7)

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

(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:

GC_IN_GGGA(.(T416, T417), T418, T419, T421) → GC_IN_GGGA(T417, T416, .(T418, T419), T421)

The TRS R consists of the following rules:

fA_in_gga([], [], []) → fA_out_gga([], [], [])
fA_in_gga(.(T21, T22), [], T24) → U1_gga(T21, T22, T24, gB_in_gga(T22, T21, T24))
gB_in_gga([], T31, .(T31, [])) → gB_out_gga([], T31, .(T31, []))
gB_in_gga(.(T57, []), T58, .(T57, .(T58, []))) → gB_out_gga(.(T57, []), T58, .(T57, .(T58, [])))
gB_in_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, [])))) → gB_out_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, []))))
gB_in_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, []))))) → gB_out_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, [])))))
gB_in_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, [])))))) → gB_out_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, []))))))
gB_in_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, []))))))) → gB_out_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, [])))))))
gB_in_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, [])))))))) → gB_out_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, []))))))))
gB_in_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374) → U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_in_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374))
gC_in_ggga([], T404, T405, .(T404, T405)) → gC_out_ggga([], T404, T405, .(T404, T405))
gC_in_ggga(.(T416, T417), T418, T419, T421) → U4_ggga(T416, T417, T418, T419, T421, gC_in_ggga(T417, T416, .(T418, T419), T421))
U4_ggga(T416, T417, T418, T419, T421, gC_out_ggga(T417, T416, .(T418, T419), T421)) → gC_out_ggga(.(T416, T417), T418, T419, T421)
U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_out_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374)) → gB_out_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374)
U1_gga(T21, T22, T24, gB_out_gga(T22, T21, T24)) → fA_out_gga(.(T21, T22), [], T24)
fA_in_gga(.(T472, T473), .(T471, []), T475) → U2_gga(T472, T473, T471, T475, gB_in_gga(.(T472, T473), T471, T475))
U2_gga(T472, T473, T471, T475, gB_out_gga(.(T472, T473), T471, T475)) → fA_out_gga(.(T472, T473), .(T471, []), T475)
fA_in_gga(.(T489, T490), .(T488, .(T491, T492)), T494) → U3_gga(T489, T490, T488, T491, T492, T494, fA_in_gga(.(T491, .(T488, .(T489, T490))), T492, T494))
U3_gga(T489, T490, T488, T491, T492, T494, fA_out_gga(.(T491, .(T488, .(T489, T490))), T492, T494)) → fA_out_gga(.(T489, T490), .(T488, .(T491, T492)), T494)

The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3)  =  fA_in_gga(x1, x2)
[]  =  []
fA_out_gga(x1, x2, x3)  =  fA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
gB_in_gga(x1, x2, x3)  =  gB_in_gga(x1, x2)
gB_out_gga(x1, x2, x3)  =  gB_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)  =  U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
gC_in_ggga(x1, x2, x3, x4)  =  gC_in_ggga(x1, x2, x3)
gC_out_ggga(x1, x2, x3, x4)  =  gC_out_ggga(x1, x2, x3, x4)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gga(x1, x2, x3, x4, x5, x7)
GC_IN_GGGA(x1, x2, x3, x4)  =  GC_IN_GGGA(x1, x2, x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

GC_IN_GGGA(.(T416, T417), T418, T419, T421) → GC_IN_GGGA(T417, T416, .(T418, T419), T421)

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

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:

GC_IN_GGGA(.(T416, T417), T418, T419) → GC_IN_GGGA(T417, T416, .(T418, T419))

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:

• GC_IN_GGGA(.(T416, T417), T418, T419) → GC_IN_GGGA(T417, T416, .(T418, T419))
  The graph contains the following edges 1 > 1, 1 > 2

(13) YES

(14) Obligation:

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

FA_IN_GGA(.(T489, T490), .(T488, .(T491, T492)), T494) → FA_IN_GGA(.(T491, .(T488, .(T489, T490))), T492, T494)

The TRS R consists of the following rules:

fA_in_gga([], [], []) → fA_out_gga([], [], [])
fA_in_gga(.(T21, T22), [], T24) → U1_gga(T21, T22, T24, gB_in_gga(T22, T21, T24))
gB_in_gga([], T31, .(T31, [])) → gB_out_gga([], T31, .(T31, []))
gB_in_gga(.(T57, []), T58, .(T57, .(T58, []))) → gB_out_gga(.(T57, []), T58, .(T57, .(T58, [])))
gB_in_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, [])))) → gB_out_gga(.(T94, .(T93, [])), T95, .(T93, .(T94, .(T95, []))))
gB_in_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, []))))) → gB_out_gga(.(T141, .(T140, .(T139, []))), T142, .(T139, .(T140, .(T141, .(T142, [])))))
gB_in_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, [])))))) → gB_out_gga(.(T198, .(T197, .(T196, .(T195, [])))), T199, .(T195, .(T196, .(T197, .(T198, .(T199, []))))))
gB_in_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, []))))))) → gB_out_gga(.(T265, .(T264, .(T263, .(T262, .(T261, []))))), T266, .(T261, .(T262, .(T263, .(T264, .(T265, .(T266, [])))))))
gB_in_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, [])))))))) → gB_out_gga(.(T342, .(T341, .(T340, .(T339, .(T338, .(T337, [])))))), T343, .(T337, .(T338, .(T339, .(T340, .(T341, .(T342, .(T343, []))))))))
gB_in_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374) → U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_in_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374))
gC_in_ggga([], T404, T405, .(T404, T405)) → gC_out_ggga([], T404, T405, .(T404, T405))
gC_in_ggga(.(T416, T417), T418, T419, T421) → U4_ggga(T416, T417, T418, T419, T421, gC_in_ggga(T417, T416, .(T418, T419), T421))
U4_ggga(T416, T417, T418, T419, T421, gC_out_ggga(T417, T416, .(T418, T419), T421)) → gC_out_ggga(.(T416, T417), T418, T419, T421)
U5_gga(T371, T370, T369, T368, T367, T366, T364, T365, T372, T374, gC_out_ggga(T365, T364, .(T366, .(T367, .(T368, .(T369, .(T370, .(T371, .(T372, []))))))), T374)) → gB_out_gga(.(T371, .(T370, .(T369, .(T368, .(T367, .(T366, .(T364, T365))))))), T372, T374)
U1_gga(T21, T22, T24, gB_out_gga(T22, T21, T24)) → fA_out_gga(.(T21, T22), [], T24)
fA_in_gga(.(T472, T473), .(T471, []), T475) → U2_gga(T472, T473, T471, T475, gB_in_gga(.(T472, T473), T471, T475))
U2_gga(T472, T473, T471, T475, gB_out_gga(.(T472, T473), T471, T475)) → fA_out_gga(.(T472, T473), .(T471, []), T475)
fA_in_gga(.(T489, T490), .(T488, .(T491, T492)), T494) → U3_gga(T489, T490, T488, T491, T492, T494, fA_in_gga(.(T491, .(T488, .(T489, T490))), T492, T494))
U3_gga(T489, T490, T488, T491, T492, T494, fA_out_gga(.(T491, .(T488, .(T489, T490))), T492, T494)) → fA_out_gga(.(T489, T490), .(T488, .(T491, T492)), T494)

The argument filtering Pi contains the following mapping:
fA_in_gga(x1, x2, x3)  =  fA_in_gga(x1, x2)
[]  =  []
fA_out_gga(x1, x2, x3)  =  fA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
gB_in_gga(x1, x2, x3)  =  gB_in_gga(x1, x2)
gB_out_gga(x1, x2, x3)  =  gB_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)  =  U5_gga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
gC_in_ggga(x1, x2, x3, x4)  =  gC_in_ggga(x1, x2, x3)
gC_out_ggga(x1, x2, x3, x4)  =  gC_out_ggga(x1, x2, x3, x4)
U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x1, x2, x3, x4, x6)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x1, x2, x3, x5)
U3_gga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gga(x1, x2, x3, x4, x5, x7)
FA_IN_GGA(x1, x2, x3)  =  FA_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:

FA_IN_GGA(.(T489, T490), .(T488, .(T491, T492)), T494) → FA_IN_GGA(.(T491, .(T488, .(T489, T490))), T492, T494)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
FA_IN_GGA(x1, x2, x3)  =  FA_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:

FA_IN_GGA(.(T489, T490), .(T488, .(T491, T492))) → FA_IN_GGA(.(T491, .(T488, .(T489, T490))), T492)

R is empty.
Q is empty.
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:

• FA_IN_GGA(.(T489, T490), .(T488, .(T491, T492))) → FA_IN_GGA(.(T491, .(T488, .(T489, T490))), T492)
  The graph contains the following edges 2 > 2

(20) YES