Left Termination proof of /tmp/tmp67ZEqZ/reverse-bf.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 253 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 10 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 10 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 0 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

reverse(X1s, X2s) :- reverse(X1s, [], X2s).
reverse([], Xs, Xs).
reverse(.(X, X1s), X2s, Ys) :- reverse(X1s, .(X, X2s), Ys).

Query: reverse(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: reverse(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: reverse(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: reverse(T5, [], T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: reverse(T5, [], T7), SCOPE: 2, CLAUSE: 1 | reverse(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: reverse(T5, [], T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: reverse(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: reverse(T15, .(T14, []), T17)\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: reverse(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 1 | reverse(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: reverse(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 1\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: reverse(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: reverse(T34, .(T33, .(T35, [])), T37)\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: reverse(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 1 | reverse(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 2\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: reverse(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 1\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: reverse(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 2\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: reverse(T63, .(T62, .(T64, .(T65, []))), T67)\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: reverse(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 1 | reverse(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 2\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: reverse(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 1\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: reverse(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 2\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: reverse(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107)\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: reverse(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 1 | reverse(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 2\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: reverse(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 1\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: reverse(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 2\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: reverse(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157)\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: reverse(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 1 | reverse(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 2\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: reverse(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 1\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: reverse(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 2\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", 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];
156 [label="156: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: reverse(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217)\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: reverse(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 1 | reverse(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 2\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: reverse(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 1\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: reverse(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 2\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="183: reverse(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287)\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
184 [label="184: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
190 [label="190: reverse(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 1 | reverse(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 2\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="191: reverse(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 1\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: reverse(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 2\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="203: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
209 [label="209: reverse(T358, .(T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, [])))))))), T367)\n\nT357 is ground\nT358 is ground\nT359 is ground\nT360 is ground\nT361 is ground\nT362 is ground\nT363 is ground\nT364 is ground\nT365 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
210 [label="210: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="B: reverse(T358, .(T357, T384), T367)\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
217 [label="217: reverse(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 1 | reverse(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 2\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
218 [label="218: reverse(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 1\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
219 [label="219: reverse(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 2\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
220 [label="220: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
227 [label="227: reverse(T410, .(T409, .(T411, T412)), T414)\n\nT409 is ground\nT410 is ground\nT411 is ground\nT412 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="228: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
229 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 229 [arrowhead = none ];
229 -> 4;

230 [label="ONLY EVAL with clause\nreverse(X3, X4) :- reverse(X3, [], X4).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 230 [arrowhead = none ];
230 -> 7;

231 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 231 [arrowhead = none ];
231 -> 9;

232 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 232 [arrowhead = none ];
232 -> 10;

233 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 233 [arrowhead = none ];
233 -> 11;

234 [label="EVAL with clause\nreverse([], X9, X9).\nand substitutionT5 -> [],\nX9 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 234 [arrowhead = none ];
234 -> 12;

235 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 235 [arrowhead = none ];
235 -> 14;

236 [label="EVAL with clause\nreverse(.(X18, X19), X20, X21) :- reverse(X19, .(X18, X20), X21).\nand substitutionX18 -> T14,\nX19 -> T15,\nT5 -> .(T14, T15),\nX20 -> [],\nT7 -> T17,\nX21 -> T17,\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 236 [arrowhead = none ];
236 -> 22;

237 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 237 [arrowhead = none ];
237 -> 23;

238 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 238 [arrowhead = none ];
238 -> 16;

239 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 239 [arrowhead = none ];
239 -> 28;

240 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 240 [arrowhead = none ];
240 -> 29;

241 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 241 [arrowhead = none ];
241 -> 30;

242 [label="EVAL with clause\nreverse([], X28, X28).\nand substitutionT15 -> [],\nT14 -> T24,\nX28 -> .(T24, []),\nT17 -> .(T24, [])", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 242 [arrowhead = none ];
242 -> 31;

243 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 243 [arrowhead = none ];
243 -> 32;

244 [label="EVAL with clause\nreverse(.(X37, X38), X39, X40) :- reverse(X38, .(X37, X39), X40).\nand substitutionX37 -> T33,\nX38 -> T34,\nT15 -> .(T33, T34),\nT14 -> T35,\nX39 -> .(T35, []),\nT17 -> T37,\nX40 -> T37,\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 244 [arrowhead = none ];
244 -> 42;

245 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 245 [arrowhead = none ];
245 -> 43;

246 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 246 [arrowhead = none ];
246 -> 33;

247 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 247 [arrowhead = none ];
247 -> 48;

248 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 248 [arrowhead = none ];
248 -> 53;

249 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 249 [arrowhead = none ];
249 -> 55;

250 [label="EVAL with clause\nreverse([], X47, X47).\nand substitutionT34 -> [],\nT33 -> T50,\nT35 -> T51,\nX47 -> .(T50, .(T51, [])),\nT37 -> .(T50, .(T51, []))", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 250 [arrowhead = none ];
250 -> 64;

251 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 251 [arrowhead = none ];
251 -> 65;

252 [label="EVAL with clause\nreverse(.(X56, X57), X58, X59) :- reverse(X57, .(X56, X58), X59).\nand substitutionX56 -> T62,\nX57 -> T63,\nT34 -> .(T62, T63),\nT33 -> T64,\nT35 -> T65,\nX58 -> .(T64, .(T65, [])),\nT37 -> T67,\nX59 -> T67,\nT66 -> T67", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 252 [arrowhead = none ];
252 -> 78;

253 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 253 [arrowhead = none ];
253 -> 80;

254 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 254 [arrowhead = none ];
254 -> 66;

255 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
78 -> 255 [arrowhead = none ];
255 -> 82;

256 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 256 [arrowhead = none ];
256 -> 95;

257 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
82 -> 257 [arrowhead = none ];
257 -> 96;

258 [label="EVAL with clause\nreverse([], X66, X66).\nand substitutionT63 -> [],\nT62 -> T86,\nT64 -> T87,\nT65 -> T88,\nX66 -> .(T86, .(T87, .(T88, []))),\nT67 -> .(T86, .(T87, .(T88, [])))", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 258 [arrowhead = none ];
258 -> 98;

259 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 259 [arrowhead = none ];
259 -> 100;

260 [label="EVAL with clause\nreverse(.(X75, X76), X77, X78) :- reverse(X76, .(X75, X77), X78).\nand substitutionX75 -> T101,\nX76 -> T102,\nT63 -> .(T101, T102),\nT62 -> T103,\nT64 -> T104,\nT65 -> T105,\nX77 -> .(T103, .(T104, .(T105, []))),\nT67 -> T107,\nX78 -> T107,\nT106 -> T107", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 260 [arrowhead = none ];
260 -> 105;

261 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 261 [arrowhead = none ];
261 -> 106;

262 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
98 -> 262 [arrowhead = none ];
262 -> 101;

263 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
105 -> 263 [arrowhead = none ];
263 -> 115;

264 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
115 -> 264 [arrowhead = none ];
264 -> 117;

265 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
115 -> 265 [arrowhead = none ];
265 -> 118;

266 [label="EVAL with clause\nreverse([], X85, X85).\nand substitutionT102 -> [],\nT101 -> T132,\nT103 -> T133,\nT104 -> T134,\nT105 -> T135,\nX85 -> .(T132, .(T133, .(T134, .(T135, [])))),\nT107 -> .(T132, .(T133, .(T134, .(T135, []))))", fontsize=14, style = filled, fillcolor = lightblue];
117 -> 266 [arrowhead = none ];
266 -> 122;

267 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
117 -> 267 [arrowhead = none ];
267 -> 125;

268 [label="EVAL with clause\nreverse(.(X94, X95), X96, X97) :- reverse(X95, .(X94, X96), X97).\nand substitutionX94 -> T150,\nX95 -> T151,\nT102 -> .(T150, T151),\nT101 -> T152,\nT103 -> T153,\nT104 -> T154,\nT105 -> T155,\nX96 -> .(T152, .(T153, .(T154, .(T155, [])))),\nT107 -> T157,\nX97 -> T157,\nT156 -> T157", fontsize=14, style = filled, fillcolor = lightblue];
118 -> 268 [arrowhead = none ];
268 -> 129;

269 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
118 -> 269 [arrowhead = none ];
269 -> 131;

270 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
122 -> 270 [arrowhead = none ];
270 -> 127;

271 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
129 -> 271 [arrowhead = none ];
271 -> 144;

272 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
144 -> 272 [arrowhead = none ];
272 -> 145;

273 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
144 -> 273 [arrowhead = none ];
273 -> 146;

274 [label="EVAL with clause\nreverse([], X104, X104).\nand substitutionT151 -> [],\nT150 -> T188,\nT152 -> T189,\nT153 -> T190,\nT154 -> T191,\nT155 -> T192,\nX104 -> .(T188, .(T189, .(T190, .(T191, .(T192, []))))),\nT157 -> .(T188, .(T189, .(T190, .(T191, .(T192, [])))))", fontsize=14, style = filled, fillcolor = lightblue];
145 -> 274 [arrowhead = none ];
274 -> 153;

275 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
145 -> 275 [arrowhead = none ];
275 -> 154;

276 [label="EVAL with clause\nreverse(.(X113, X114), X115, X116) :- reverse(X114, .(X113, X115), X116).\nand substitutionX113 -> T209,\nX114 -> T210,\nT151 -> .(T209, T210),\nT150 -> T211,\nT152 -> T212,\nT153 -> T213,\nT154 -> T214,\nT155 -> T215,\nX115 -> .(T211, .(T212, .(T213, .(T214, .(T215, []))))),\nT157 -> T217,\nX116 -> T217,\nT216 -> T217", fontsize=14, style = filled, fillcolor = lightblue];
146 -> 276 [arrowhead = none ];
276 -> 162;

277 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
146 -> 277 [arrowhead = none ];
277 -> 163;

278 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
153 -> 278 [arrowhead = none ];
278 -> 156;

279 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
162 -> 279 [arrowhead = none ];
279 -> 166;

280 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
166 -> 280 [arrowhead = none ];
280 -> 170;

281 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
166 -> 281 [arrowhead = none ];
281 -> 171;

282 [label="EVAL with clause\nreverse([], X123, X123).\nand substitutionT210 -> [],\nT209 -> T254,\nT211 -> T255,\nT212 -> T256,\nT213 -> T257,\nT214 -> T258,\nT215 -> T259,\nX123 -> .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))),\nT217 -> .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))", fontsize=14, style = filled, fillcolor = lightblue];
170 -> 282 [arrowhead = none ];
282 -> 174;

283 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
170 -> 283 [arrowhead = none ];
283 -> 175;

284 [label="EVAL with clause\nreverse(.(X132, X133), X134, X135) :- reverse(X133, .(X132, X134), X135).\nand substitutionX132 -> T278,\nX133 -> T279,\nT210 -> .(T278, T279),\nT209 -> T280,\nT211 -> T281,\nT212 -> T282,\nT213 -> T283,\nT214 -> T284,\nT215 -> T285,\nX134 -> .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, [])))))),\nT217 -> T287,\nX135 -> T287,\nT286 -> T287", fontsize=14, style = filled, fillcolor = lightblue];
171 -> 284 [arrowhead = none ];
284 -> 183;

285 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
171 -> 285 [arrowhead = none ];
285 -> 184;

286 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
174 -> 286 [arrowhead = none ];
286 -> 176;

287 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
183 -> 287 [arrowhead = none ];
287 -> 190;

288 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
190 -> 288 [arrowhead = none ];
288 -> 191;

289 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
190 -> 289 [arrowhead = none ];
289 -> 192;

290 [label="EVAL with clause\nreverse([], X142, X142).\nand substitutionT279 -> [],\nT278 -> T330,\nT280 -> T331,\nT281 -> T332,\nT282 -> T333,\nT283 -> T334,\nT284 -> T335,\nT285 -> T336,\nX142 -> .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))),\nT287 -> .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
191 -> 290 [arrowhead = none ];
290 -> 202;

291 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
191 -> 291 [arrowhead = none ];
291 -> 203;

292 [label="EVAL with clause\nreverse(.(X151, X152), X153, X154) :- reverse(X152, .(X151, X153), X154).\nand substitutionX151 -> T357,\nX152 -> T358,\nT279 -> .(T357, T358),\nT278 -> T359,\nT280 -> T360,\nT281 -> T361,\nT282 -> T362,\nT283 -> T363,\nT284 -> T364,\nT285 -> T365,\nX153 -> .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))),\nT287 -> T367,\nX154 -> T367,\nT366 -> T367", fontsize=14, style = filled, fillcolor = lightblue];
192 -> 292 [arrowhead = none ];
292 -> 209;

293 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
192 -> 293 [arrowhead = none ];
293 -> 210;

294 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
202 -> 294 [arrowhead = none ];
294 -> 204;

295 [label="GENERALIZATION\nT384 <-- .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, [])))))))\n\nNew Knowledge:\nT384 is ground", fontsize=14, style = filled, fillcolor = violet];
209 -> 295 [arrowhead = none ];
295 -> 216;

296 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
216 -> 296 [arrowhead = none ];
296 -> 217;

297 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
217 -> 297 [arrowhead = none ];
297 -> 218;

298 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
217 -> 298 [arrowhead = none ];
298 -> 219;

299 [label="EVAL with clause\nreverse([], X163, X163).\nand substitutionT358 -> [],\nT357 -> T397,\nT384 -> T398,\nX163 -> .(T397, T398),\nT367 -> .(T397, T398)", fontsize=14, style = filled, fillcolor = lightblue];
218 -> 299 [arrowhead = none ];
299 -> 220;

300 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
218 -> 300 [arrowhead = none ];
300 -> 221;

301 [label="EVAL with clause\nreverse(.(X172, X173), X174, X175) :- reverse(X173, .(X172, X174), X175).\nand substitutionX172 -> T409,\nX173 -> T410,\nT358 -> .(T409, T410),\nT357 -> T411,\nT384 -> T412,\nX174 -> .(T411, T412),\nT367 -> T414,\nX175 -> T414,\nT413 -> T414", fontsize=14, style = filled, fillcolor = lightblue];
219 -> 301 [arrowhead = none ];
301 -> 227;

302 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
219 -> 302 [arrowhead = none ];
302 -> 228;

303 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
220 -> 303 [arrowhead = none ];
303 -> 222;

304 [label="INSTANCE with matching:\nT358 -> T410\nT357 -> T409\nT384 -> .(T411, T412)\nT367 -> T414", fontsize=14, style = filled, fillcolor = lightgrey];
227 -> 304 [arrowhead = none , style=dashed];
304 -> 216[style=dashed];

}

(2) Obligation:

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

reverseA_in_ga([], []) → reverseA_out_ga([], [])
reverseA_in_ga(.(T24, []), .(T24, [])) → reverseA_out_ga(.(T24, []), .(T24, []))
reverseA_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → reverseA_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
reverseA_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → reverseA_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
reverseA_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → reverseA_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
reverseA_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → reverseA_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
reverseA_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → reverseA_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
reverseA_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → reverseA_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
reverseA_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
reverseB_in_ggga([], T397, T398, .(T397, T398)) → reverseB_out_ggga([], T397, T398, .(T397, T398))
reverseB_in_ggga(.(T409, T410), T411, T412, T414) → U2_ggga(T409, T410, T411, T412, T414, reverseB_in_ggga(T410, T409, .(T411, T412), T414))
U2_ggga(T409, T410, T411, T412, T414, reverseB_out_ggga(T410, T409, .(T411, T412), T414)) → reverseB_out_ggga(.(T409, T410), T411, T412, T414)
U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → reverseA_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

(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:

REVERSEA_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_GA(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
REVERSEA_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → REVERSEB_IN_GGGA(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)
REVERSEB_IN_GGGA(.(T409, T410), T411, T412, T414) → U2_GGGA(T409, T410, T411, T412, T414, reverseB_in_ggga(T410, T409, .(T411, T412), T414))
REVERSEB_IN_GGGA(.(T409, T410), T411, T412, T414) → REVERSEB_IN_GGGA(T410, T409, .(T411, T412), T414)

The TRS R consists of the following rules:

reverseA_in_ga([], []) → reverseA_out_ga([], [])
reverseA_in_ga(.(T24, []), .(T24, [])) → reverseA_out_ga(.(T24, []), .(T24, []))
reverseA_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → reverseA_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
reverseA_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → reverseA_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
reverseA_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → reverseA_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
reverseA_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → reverseA_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
reverseA_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → reverseA_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
reverseA_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → reverseA_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
reverseA_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
reverseB_in_ggga([], T397, T398, .(T397, T398)) → reverseB_out_ggga([], T397, T398, .(T397, T398))
reverseB_in_ggga(.(T409, T410), T411, T412, T414) → U2_ggga(T409, T410, T411, T412, T414, reverseB_in_ggga(T410, T409, .(T411, T412), T414))
U2_ggga(T409, T410, T411, T412, T414, reverseB_out_ggga(T410, T409, .(T411, T412), T414)) → reverseB_out_ggga(.(T409, T410), T411, T412, T414)
U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → reverseA_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

The argument filtering Pi contains the following mapping:
reverseA_in_ga(x1, x2) = reverseA_in_ga(x1)
[] = []
reverseA_out_ga(x1, x2) = reverseA_out_ga(x1, x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U1_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
reverseB_in_ggga(x1, x2, x3, x4) = reverseB_in_ggga(x1, x2, x3)
reverseB_out_ggga(x1, x2, x3, x4) = reverseB_out_ggga(x1, x2, x3, x4)
U2_ggga(x1, x2, x3, x4, x5, x6) = U2_ggga(x1, x2, x3, x4, x6)
REVERSEA_IN_GA(x1, x2) = REVERSEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U1_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
REVERSEB_IN_GGGA(x1, x2, x3, x4) = REVERSEB_IN_GGGA(x1, x2, x3)
U2_GGGA(x1, x2, x3, x4, x5, x6) = U2_GGGA(x1, x2, x3, x4, x6)

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

(4) Obligation:

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

REVERSEA_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_GA(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
REVERSEA_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → REVERSEB_IN_GGGA(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)
REVERSEB_IN_GGGA(.(T409, T410), T411, T412, T414) → U2_GGGA(T409, T410, T411, T412, T414, reverseB_in_ggga(T410, T409, .(T411, T412), T414))
REVERSEB_IN_GGGA(.(T409, T410), T411, T412, T414) → REVERSEB_IN_GGGA(T410, T409, .(T411, T412), T414)

The TRS R consists of the following rules:

reverseA_in_ga([], []) → reverseA_out_ga([], [])
reverseA_in_ga(.(T24, []), .(T24, [])) → reverseA_out_ga(.(T24, []), .(T24, []))
reverseA_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → reverseA_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
reverseA_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → reverseA_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
reverseA_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → reverseA_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
reverseA_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → reverseA_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
reverseA_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → reverseA_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
reverseA_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → reverseA_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
reverseA_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
reverseB_in_ggga([], T397, T398, .(T397, T398)) → reverseB_out_ggga([], T397, T398, .(T397, T398))
reverseB_in_ggga(.(T409, T410), T411, T412, T414) → U2_ggga(T409, T410, T411, T412, T414, reverseB_in_ggga(T410, T409, .(T411, T412), T414))
U2_ggga(T409, T410, T411, T412, T414, reverseB_out_ggga(T410, T409, .(T411, T412), T414)) → reverseB_out_ggga(.(T409, T410), T411, T412, T414)
U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → reverseA_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(6) Obligation:

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

REVERSEB_IN_GGGA(.(T409, T410), T411, T412, T414) → REVERSEB_IN_GGGA(T410, T409, .(T411, T412), T414)

The TRS R consists of the following rules:

reverseA_in_ga([], []) → reverseA_out_ga([], [])
reverseA_in_ga(.(T24, []), .(T24, [])) → reverseA_out_ga(.(T24, []), .(T24, []))
reverseA_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → reverseA_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
reverseA_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → reverseA_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
reverseA_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → reverseA_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
reverseA_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → reverseA_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
reverseA_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → reverseA_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
reverseA_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → reverseA_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
reverseA_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
reverseB_in_ggga([], T397, T398, .(T397, T398)) → reverseB_out_ggga([], T397, T398, .(T397, T398))
reverseB_in_ggga(.(T409, T410), T411, T412, T414) → U2_ggga(T409, T410, T411, T412, T414, reverseB_in_ggga(T410, T409, .(T411, T412), T414))
U2_ggga(T409, T410, T411, T412, T414, reverseB_out_ggga(T410, T409, .(T411, T412), T414)) → reverseB_out_ggga(.(T409, T410), T411, T412, T414)
U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, reverseB_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → reverseA_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

The argument filtering Pi contains the following mapping:
reverseA_in_ga(x1, x2) = reverseA_in_ga(x1)
[] = []
reverseA_out_ga(x1, x2) = reverseA_out_ga(x1, x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U1_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
reverseB_in_ggga(x1, x2, x3, x4) = reverseB_in_ggga(x1, x2, x3)
reverseB_out_ggga(x1, x2, x3, x4) = reverseB_out_ggga(x1, x2, x3, x4)
U2_ggga(x1, x2, x3, x4, x5, x6) = U2_ggga(x1, x2, x3, x4, x6)
REVERSEB_IN_GGGA(x1, x2, x3, x4) = REVERSEB_IN_GGGA(x1, x2, x3)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

REVERSEB_IN_GGGA(.(T409, T410), T411, T412, T414) → REVERSEB_IN_GGGA(T410, T409, .(T411, T412), T414)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

REVERSEB_IN_GGGA(.(T409, T410), T411, T412) → REVERSEB_IN_GGGA(T410, T409, .(T411, T412))

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

(11) 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:

REVERSEB_IN_GGGA(.(T409, T410), T411, T412) → REVERSEB_IN_GGGA(T410, T409, .(T411, T412))
The graph contains the following edges 1 > 1, 1 > 2

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) PiDPToQDPProof (SOUND transformation)

(10) Obligation:

(11) QDPSizeChangeProof (EQUIVALENT transformation)

(12) YES