Left Termination proof of /tmp/tmpNJHq28/reverse.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 (⇒, 292 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 9 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 8 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 0 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 3 ms)

↳12 YES

(0) Obligation:

Clauses:

reverse(L, LR) :- revacc(L, LR, []).
revacc([], L, L).
revacc(.(EL, T), R, A) :- revacc(T, R, .(EL, A)).

Query: reverse(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: reverse(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: reverse(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: revacc(T5, T7, [])\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: revacc(T5, T7, []), SCOPE: 2, CLAUSE: 1 | revacc(T5, T7, []), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: revacc(T5, T7, []), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: revacc(T5, T7, []), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: revacc(T20, T22, .(T19, []))\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: revacc(T20, T22, .(T19, [])), SCOPE: 3, CLAUSE: 1 | revacc(T20, T22, .(T19, [])), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: revacc(T20, T22, .(T19, [])), SCOPE: 3, CLAUSE: 1\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: revacc(T20, T22, .(T19, [])), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: revacc(T46, T49, .(T45, .(T48, [])))\n\nT45 is ground\nT46 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: revacc(T46, T49, .(T45, .(T48, []))), SCOPE: 4, CLAUSE: 1 | revacc(T46, T49, .(T45, .(T48, []))), SCOPE: 4, CLAUSE: 2\n\nT45 is ground\nT46 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: revacc(T46, T49, .(T45, .(T48, []))), SCOPE: 4, CLAUSE: 1\n\nT45 is ground\nT46 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: revacc(T46, T49, .(T45, .(T48, []))), SCOPE: 4, CLAUSE: 2\n\nT45 is ground\nT46 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: revacc(T82, T86, .(T81, .(T84, .(T85, []))))\n\nT81 is ground\nT82 is ground\nT84 is ground\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: revacc(T82, T86, .(T81, .(T84, .(T85, [])))), SCOPE: 5, CLAUSE: 1 | revacc(T82, T86, .(T81, .(T84, .(T85, [])))), SCOPE: 5, CLAUSE: 2\n\nT81 is ground\nT82 is ground\nT84 is ground\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: revacc(T82, T86, .(T81, .(T84, .(T85, [])))), SCOPE: 5, CLAUSE: 1\n\nT81 is ground\nT82 is ground\nT84 is ground\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: revacc(T82, T86, .(T81, .(T84, .(T85, [])))), SCOPE: 5, CLAUSE: 2\n\nT81 is ground\nT82 is ground\nT84 is ground\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, [])))))\n\nT127 is ground\nT128 is ground\nT130 is ground\nT131 is ground\nT132 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, []))))), SCOPE: 6, CLAUSE: 1 | revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, []))))), SCOPE: 6, CLAUSE: 2\n\nT127 is ground\nT128 is ground\nT130 is ground\nT131 is ground\nT132 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, []))))), SCOPE: 6, CLAUSE: 1\n\nT127 is ground\nT128 is ground\nT130 is ground\nT131 is ground\nT132 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, []))))), SCOPE: 6, CLAUSE: 2\n\nT127 is ground\nT128 is ground\nT130 is ground\nT131 is ground\nT132 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, []))))))\n\nT183 is ground\nT184 is ground\nT186 is ground\nT187 is ground\nT188 is ground\nT189 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, [])))))), SCOPE: 7, CLAUSE: 1 | revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, [])))))), SCOPE: 7, CLAUSE: 2\n\nT183 is ground\nT184 is ground\nT186 is ground\nT187 is ground\nT188 is ground\nT189 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
147 [label="147: revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, [])))))), SCOPE: 7, CLAUSE: 1\n\nT183 is ground\nT184 is ground\nT186 is ground\nT187 is ground\nT188 is ground\nT189 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, [])))))), SCOPE: 7, CLAUSE: 2\n\nT183 is ground\nT184 is ground\nT186 is ground\nT187 is ground\nT188 is ground\nT189 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];
155 [label="155: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, [])))))))\n\nT249 is ground\nT250 is ground\nT252 is ground\nT253 is ground\nT254 is ground\nT255 is ground\nT256 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, []))))))), SCOPE: 8, CLAUSE: 1 | revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, []))))))), SCOPE: 8, CLAUSE: 2\n\nT249 is ground\nT250 is ground\nT252 is ground\nT253 is ground\nT254 is ground\nT255 is ground\nT256 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, []))))))), SCOPE: 8, CLAUSE: 1\n\nT249 is ground\nT250 is ground\nT252 is ground\nT253 is ground\nT254 is ground\nT255 is ground\nT256 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, []))))))), SCOPE: 8, CLAUSE: 2\n\nT249 is ground\nT250 is ground\nT252 is ground\nT253 is ground\nT254 is ground\nT255 is ground\nT256 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
178 [label="178: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
182 [label="182: revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, []))))))))\n\nT325 is ground\nT326 is ground\nT328 is ground\nT329 is ground\nT330 is ground\nT331 is ground\nT332 is ground\nT333 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="183: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
187 [label="187: revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, [])))))))), SCOPE: 9, CLAUSE: 1 | revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, [])))))))), SCOPE: 9, CLAUSE: 2\n\nT325 is ground\nT326 is ground\nT328 is ground\nT329 is ground\nT330 is ground\nT331 is ground\nT332 is ground\nT333 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
188 [label="188: revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, [])))))))), SCOPE: 9, CLAUSE: 1\n\nT325 is ground\nT326 is ground\nT328 is ground\nT329 is ground\nT330 is ground\nT331 is ground\nT332 is ground\nT333 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
189 [label="189: revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, [])))))))), SCOPE: 9, CLAUSE: 2\n\nT325 is ground\nT326 is ground\nT328 is ground\nT329 is ground\nT330 is ground\nT331 is ground\nT332 is ground\nT333 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
195 [label="195: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
196 [label="196: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="197: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: revacc(T412, T421, .(T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))\n\nT411 is ground\nT412 is ground\nT414 is ground\nT415 is ground\nT416 is ground\nT417 is ground\nT418 is ground\nT419 is ground\nT420 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="203: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
207 [label="B: revacc(T412, T421, .(T411, T440))\n\nT411 is ground\nT412 is ground\nT440 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="212: revacc(T412, T421, .(T411, T440)), SCOPE: 10, CLAUSE: 1 | revacc(T412, T421, .(T411, T440)), SCOPE: 10, CLAUSE: 2\n\nT411 is ground\nT412 is ground\nT440 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="213: revacc(T412, T421, .(T411, T440)), SCOPE: 10, CLAUSE: 1\n\nT411 is ground\nT412 is ground\nT440 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
214 [label="214: revacc(T412, T421, .(T411, T440)), SCOPE: 10, CLAUSE: 2\n\nT411 is ground\nT412 is ground\nT440 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="216: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
218 [label="218: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
219 [label="219: revacc(T473, T477, .(T472, .(T475, T476)))\n\nT472 is ground\nT473 is ground\nT475 is ground\nT476 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
220 [label="220: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 221 [arrowhead = none ];
221 -> 5;

222 [label="ONLY EVAL with clause\nreverse(X3, X4) :- revacc(X3, X4, []).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 222 [arrowhead = none ];
222 -> 7;

223 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 223 [arrowhead = none ];
223 -> 10;

224 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 224 [arrowhead = none ];
224 -> 11;

225 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 225 [arrowhead = none ];
225 -> 12;

226 [label="EVAL with clause\nrevacc([], X9, X9).\nand substitutionT5 -> [],\nT7 -> [],\nX9 -> [],\nT12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 226 [arrowhead = none ];
226 -> 17;

227 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 227 [arrowhead = none ];
227 -> 18;

228 [label="EVAL with clause\nrevacc(.(X18, X19), X20, X21) :- revacc(X19, X20, .(X18, X21)).\nand substitutionX18 -> T19,\nX19 -> T20,\nT5 -> .(T19, T20),\nT7 -> T22,\nX20 -> T22,\nX21 -> [],\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 228 [arrowhead = none ];
228 -> 32;

229 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 229 [arrowhead = none ];
229 -> 33;

230 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 230 [arrowhead = none ];
230 -> 24;

231 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 231 [arrowhead = none ];
231 -> 46;

232 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 232 [arrowhead = none ];
232 -> 47;

233 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 233 [arrowhead = none ];
233 -> 48;

234 [label="EVAL with clause\nrevacc([], X28, X28).\nand substitutionT20 -> [],\nT22 -> .(T36, []),\nX28 -> .(T36, []),\nT19 -> T36,\nT35 -> .(T36, [])", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 234 [arrowhead = none ];
234 -> 49;

235 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 235 [arrowhead = none ];
235 -> 50;

236 [label="EVAL with clause\nrevacc(.(X37, X38), X39, X40) :- revacc(X38, X39, .(X37, X40)).\nand substitutionX37 -> T45,\nX38 -> T46,\nT20 -> .(T45, T46),\nT22 -> T49,\nX39 -> T49,\nT19 -> T48,\nX40 -> .(T48, []),\nT47 -> T49", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 236 [arrowhead = none ];
236 -> 55;

237 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 237 [arrowhead = none ];
237 -> 56;

238 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 238 [arrowhead = none ];
238 -> 51;

239 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 239 [arrowhead = none ];
239 -> 67;

240 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
67 -> 240 [arrowhead = none ];
240 -> 68;

241 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
67 -> 241 [arrowhead = none ];
241 -> 69;

242 [label="EVAL with clause\nrevacc([], X47, X47).\nand substitutionT46 -> [],\nT49 -> .(T69, .(T70, [])),\nX47 -> .(T69, .(T70, [])),\nT45 -> T69,\nT48 -> T70,\nT68 -> .(T69, .(T70, []))", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 242 [arrowhead = none ];
242 -> 75;

243 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 243 [arrowhead = none ];
243 -> 76;

244 [label="EVAL with clause\nrevacc(.(X56, X57), X58, X59) :- revacc(X57, X58, .(X56, X59)).\nand substitutionX56 -> T81,\nX57 -> T82,\nT46 -> .(T81, T82),\nT49 -> T86,\nX58 -> T86,\nT45 -> T84,\nT48 -> T85,\nX59 -> .(T84, .(T85, [])),\nT83 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 244 [arrowhead = none ];
244 -> 92;

245 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 245 [arrowhead = none ];
245 -> 93;

246 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 246 [arrowhead = none ];
246 -> 77;

247 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
92 -> 247 [arrowhead = none ];
247 -> 96;

248 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
96 -> 248 [arrowhead = none ];
248 -> 108;

249 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
96 -> 249 [arrowhead = none ];
249 -> 109;

250 [label="EVAL with clause\nrevacc([], X66, X66).\nand substitutionT82 -> [],\nT86 -> .(T112, .(T113, .(T114, []))),\nX66 -> .(T112, .(T113, .(T114, []))),\nT81 -> T112,\nT84 -> T113,\nT85 -> T114,\nT111 -> .(T112, .(T113, .(T114, [])))", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 250 [arrowhead = none ];
250 -> 113;

251 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 251 [arrowhead = none ];
251 -> 114;

252 [label="EVAL with clause\nrevacc(.(X75, X76), X77, X78) :- revacc(X76, X77, .(X75, X78)).\nand substitutionX75 -> T127,\nX76 -> T128,\nT82 -> .(T127, T128),\nT86 -> T133,\nX77 -> T133,\nT81 -> T130,\nT84 -> T131,\nT85 -> T132,\nX78 -> .(T130, .(T131, .(T132, []))),\nT129 -> T133", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 252 [arrowhead = none ];
252 -> 121;

253 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 253 [arrowhead = none ];
253 -> 122;

254 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 254 [arrowhead = none ];
254 -> 115;

255 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
121 -> 255 [arrowhead = none ];
255 -> 126;

256 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
126 -> 256 [arrowhead = none ];
256 -> 127;

257 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
126 -> 257 [arrowhead = none ];
257 -> 128;

258 [label="EVAL with clause\nrevacc([], X85, X85).\nand substitutionT128 -> [],\nT133 -> .(T165, .(T166, .(T167, .(T168, [])))),\nX85 -> .(T165, .(T166, .(T167, .(T168, [])))),\nT127 -> T165,\nT130 -> T166,\nT131 -> T167,\nT132 -> T168,\nT164 -> .(T165, .(T166, .(T167, .(T168, []))))", fontsize=14, style = filled, fillcolor = lightblue];
127 -> 258 [arrowhead = none ];
258 -> 134;

259 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
127 -> 259 [arrowhead = none ];
259 -> 135;

260 [label="EVAL with clause\nrevacc(.(X94, X95), X96, X97) :- revacc(X95, X96, .(X94, X97)).\nand substitutionX94 -> T183,\nX95 -> T184,\nT128 -> .(T183, T184),\nT133 -> T190,\nX96 -> T190,\nT127 -> T186,\nT130 -> T187,\nT131 -> T188,\nT132 -> T189,\nX97 -> .(T186, .(T187, .(T188, .(T189, [])))),\nT185 -> T190", fontsize=14, style = filled, fillcolor = lightblue];
128 -> 260 [arrowhead = none ];
260 -> 140;

261 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 261 [arrowhead = none ];
261 -> 141;

262 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
134 -> 262 [arrowhead = none ];
262 -> 136;

263 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
140 -> 263 [arrowhead = none ];
263 -> 146;

264 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
146 -> 264 [arrowhead = none ];
264 -> 147;

265 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
146 -> 265 [arrowhead = none ];
265 -> 148;

266 [label="EVAL with clause\nrevacc([], X104, X104).\nand substitutionT184 -> [],\nT190 -> .(T228, .(T229, .(T230, .(T231, .(T232, []))))),\nX104 -> .(T228, .(T229, .(T230, .(T231, .(T232, []))))),\nT183 -> T228,\nT186 -> T229,\nT187 -> T230,\nT188 -> T231,\nT189 -> T232,\nT227 -> .(T228, .(T229, .(T230, .(T231, .(T232, [])))))", fontsize=14, style = filled, fillcolor = lightblue];
147 -> 266 [arrowhead = none ];
266 -> 153;

267 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
147 -> 267 [arrowhead = none ];
267 -> 154;

268 [label="EVAL with clause\nrevacc(.(X113, X114), X115, X116) :- revacc(X114, X115, .(X113, X116)).\nand substitutionX113 -> T249,\nX114 -> T250,\nT184 -> .(T249, T250),\nT190 -> T257,\nX115 -> T257,\nT183 -> T252,\nT186 -> T253,\nT187 -> T254,\nT188 -> T255,\nT189 -> T256,\nX116 -> .(T252, .(T253, .(T254, .(T255, .(T256, []))))),\nT251 -> T257", fontsize=14, style = filled, fillcolor = lightblue];
148 -> 268 [arrowhead = none ];
268 -> 166;

269 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
148 -> 269 [arrowhead = none ];
269 -> 167;

270 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
153 -> 270 [arrowhead = none ];
270 -> 155;

271 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
166 -> 271 [arrowhead = none ];
271 -> 170;

272 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
170 -> 272 [arrowhead = none ];
272 -> 172;

273 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
170 -> 273 [arrowhead = none ];
273 -> 173;

274 [label="EVAL with clause\nrevacc([], X123, X123).\nand substitutionT250 -> [],\nT257 -> .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))),\nX123 -> .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))),\nT249 -> T301,\nT252 -> T302,\nT253 -> T303,\nT254 -> T304,\nT255 -> T305,\nT256 -> T306,\nT300 -> .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))", fontsize=14, style = filled, fillcolor = lightblue];
172 -> 274 [arrowhead = none ];
274 -> 176;

275 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
172 -> 275 [arrowhead = none ];
275 -> 177;

276 [label="EVAL with clause\nrevacc(.(X132, X133), X134, X135) :- revacc(X133, X134, .(X132, X135)).\nand substitutionX132 -> T325,\nX133 -> T326,\nT250 -> .(T325, T326),\nT257 -> T334,\nX134 -> T334,\nT249 -> T328,\nT252 -> T329,\nT253 -> T330,\nT254 -> T331,\nT255 -> T332,\nT256 -> T333,\nX135 -> .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, [])))))),\nT327 -> T334", fontsize=14, style = filled, fillcolor = lightblue];
173 -> 276 [arrowhead = none ];
276 -> 182;

277 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
173 -> 277 [arrowhead = none ];
277 -> 183;

278 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
176 -> 278 [arrowhead = none ];
278 -> 178;

279 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
182 -> 279 [arrowhead = none ];
279 -> 187;

280 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
187 -> 280 [arrowhead = none ];
280 -> 188;

281 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
187 -> 281 [arrowhead = none ];
281 -> 189;

282 [label="EVAL with clause\nrevacc([], X142, X142).\nand substitutionT326 -> [],\nT334 -> .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))),\nX142 -> .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))),\nT325 -> T384,\nT328 -> T385,\nT329 -> T386,\nT330 -> T387,\nT331 -> T388,\nT332 -> T389,\nT333 -> T390,\nT383 -> .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
188 -> 282 [arrowhead = none ];
282 -> 195;

283 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
188 -> 283 [arrowhead = none ];
283 -> 196;

284 [label="EVAL with clause\nrevacc(.(X151, X152), X153, X154) :- revacc(X152, X153, .(X151, X154)).\nand substitutionX151 -> T411,\nX152 -> T412,\nT326 -> .(T411, T412),\nT334 -> T421,\nX153 -> T421,\nT325 -> T414,\nT328 -> T415,\nT329 -> T416,\nT330 -> T417,\nT331 -> T418,\nT332 -> T419,\nT333 -> T420,\nX154 -> .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))),\nT413 -> T421", fontsize=14, style = filled, fillcolor = lightblue];
189 -> 284 [arrowhead = none ];
284 -> 200;

285 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
189 -> 285 [arrowhead = none ];
285 -> 203;

286 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
195 -> 286 [arrowhead = none ];
286 -> 197;

287 [label="GENERALIZATION\nT440 <-- .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))\n\nNew Knowledge:\nT440 is ground", fontsize=14, style = filled, fillcolor = violet];
200 -> 287 [arrowhead = none ];
287 -> 207;

288 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
207 -> 288 [arrowhead = none ];
288 -> 212;

289 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
212 -> 289 [arrowhead = none ];
289 -> 213;

290 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
212 -> 290 [arrowhead = none ];
290 -> 214;

291 [label="EVAL with clause\nrevacc([], X163, X163).\nand substitutionT412 -> [],\nT421 -> .(T460, T461),\nX163 -> .(T460, T461),\nT411 -> T460,\nT440 -> T461,\nT459 -> .(T460, T461)", fontsize=14, style = filled, fillcolor = lightblue];
213 -> 291 [arrowhead = none ];
291 -> 215;

292 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
213 -> 292 [arrowhead = none ];
292 -> 216;

293 [label="EVAL with clause\nrevacc(.(X172, X173), X174, X175) :- revacc(X173, X174, .(X172, X175)).\nand substitutionX172 -> T472,\nX173 -> T473,\nT412 -> .(T472, T473),\nT421 -> T477,\nX174 -> T477,\nT411 -> T475,\nT440 -> T476,\nX175 -> .(T475, T476),\nT474 -> T477", fontsize=14, style = filled, fillcolor = lightblue];
214 -> 293 [arrowhead = none ];
293 -> 219;

294 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
214 -> 294 [arrowhead = none ];
294 -> 220;

295 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
215 -> 295 [arrowhead = none ];
295 -> 218;

296 [label="INSTANCE with matching:\nT412 -> T473\nT421 -> T477\nT411 -> T472\nT440 -> .(T475, T476)", fontsize=14, style = filled, fillcolor = lightgrey];
219 -> 296 [arrowhead = none , style=dashed];
296 -> 207[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(.(T36, []), .(T36, [])) → reverseA_out_ga(.(T36, []), .(T36, []))
reverseA_in_ga(.(T70, .(T69, [])), .(T69, .(T70, []))) → reverseA_out_ga(.(T70, .(T69, [])), .(T69, .(T70, [])))
reverseA_in_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))) → reverseA_out_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, []))))
reverseA_in_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))) → reverseA_out_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, [])))))
reverseA_in_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))) → reverseA_out_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, []))))))
reverseA_in_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))) → reverseA_out_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))))
reverseA_in_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))) → reverseA_out_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))))
reverseA_in_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U1_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
revaccB_in_gagg([], .(T460, T461), T460, T461) → revaccB_out_gagg([], .(T460, T461), T460, T461)
revaccB_in_gagg(.(T472, T473), T477, T475, T476) → U2_gagg(T472, T473, T477, T475, T476, revaccB_in_gagg(T473, T477, T472, .(T475, T476)))
U2_gagg(T472, T473, T477, T475, T476, revaccB_out_gagg(T473, T477, T472, .(T475, T476))) → revaccB_out_gagg(.(T472, T473), T477, T475, T476)
U1_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_out_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))) → reverseA_out_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421)

(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(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U1_GA(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
REVERSEA_IN_GA(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → REVACCB_IN_GAGG(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))
REVACCB_IN_GAGG(.(T472, T473), T477, T475, T476) → U2_GAGG(T472, T473, T477, T475, T476, revaccB_in_gagg(T473, T477, T472, .(T475, T476)))
REVACCB_IN_GAGG(.(T472, T473), T477, T475, T476) → REVACCB_IN_GAGG(T473, T477, T472, .(T475, T476))

The TRS R consists of the following rules:

reverseA_in_ga([], []) → reverseA_out_ga([], [])
reverseA_in_ga(.(T36, []), .(T36, [])) → reverseA_out_ga(.(T36, []), .(T36, []))
reverseA_in_ga(.(T70, .(T69, [])), .(T69, .(T70, []))) → reverseA_out_ga(.(T70, .(T69, [])), .(T69, .(T70, [])))
reverseA_in_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))) → reverseA_out_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, []))))
reverseA_in_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))) → reverseA_out_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, [])))))
reverseA_in_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))) → reverseA_out_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, []))))))
reverseA_in_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))) → reverseA_out_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))))
reverseA_in_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))) → reverseA_out_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))))
reverseA_in_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U1_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
revaccB_in_gagg([], .(T460, T461), T460, T461) → revaccB_out_gagg([], .(T460, T461), T460, T461)
revaccB_in_gagg(.(T472, T473), T477, T475, T476) → U2_gagg(T472, T473, T477, T475, T476, revaccB_in_gagg(T473, T477, T472, .(T475, T476)))
U2_gagg(T472, T473, T477, T475, T476, revaccB_out_gagg(T473, T477, T472, .(T475, T476))) → revaccB_out_gagg(.(T472, T473), T477, T475, T476)
U1_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_out_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))) → reverseA_out_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421)

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)
revaccB_in_gagg(x1, x2, x3, x4) = revaccB_in_gagg(x1, x3, x4)
revaccB_out_gagg(x1, x2, x3, x4) = revaccB_out_gagg(x1, x2, x3, x4)
U2_gagg(x1, x2, x3, x4, x5, x6) = U2_gagg(x1, x2, x4, x5, 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)
REVACCB_IN_GAGG(x1, x2, x3, x4) = REVACCB_IN_GAGG(x1, x3, x4)
U2_GAGG(x1, x2, x3, x4, x5, x6) = U2_GAGG(x1, x2, x4, x5, 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(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U1_GA(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
REVERSEA_IN_GA(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → REVACCB_IN_GAGG(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))
REVACCB_IN_GAGG(.(T472, T473), T477, T475, T476) → U2_GAGG(T472, T473, T477, T475, T476, revaccB_in_gagg(T473, T477, T472, .(T475, T476)))
REVACCB_IN_GAGG(.(T472, T473), T477, T475, T476) → REVACCB_IN_GAGG(T473, T477, T472, .(T475, T476))

The TRS R consists of the following rules:

reverseA_in_ga([], []) → reverseA_out_ga([], [])
reverseA_in_ga(.(T36, []), .(T36, [])) → reverseA_out_ga(.(T36, []), .(T36, []))
reverseA_in_ga(.(T70, .(T69, [])), .(T69, .(T70, []))) → reverseA_out_ga(.(T70, .(T69, [])), .(T69, .(T70, [])))
reverseA_in_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))) → reverseA_out_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, []))))
reverseA_in_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))) → reverseA_out_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, [])))))
reverseA_in_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))) → reverseA_out_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, []))))))
reverseA_in_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))) → reverseA_out_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))))
reverseA_in_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))) → reverseA_out_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))))
reverseA_in_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U1_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
revaccB_in_gagg([], .(T460, T461), T460, T461) → revaccB_out_gagg([], .(T460, T461), T460, T461)
revaccB_in_gagg(.(T472, T473), T477, T475, T476) → U2_gagg(T472, T473, T477, T475, T476, revaccB_in_gagg(T473, T477, T472, .(T475, T476)))
U2_gagg(T472, T473, T477, T475, T476, revaccB_out_gagg(T473, T477, T472, .(T475, T476))) → revaccB_out_gagg(.(T472, T473), T477, T475, T476)
U1_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_out_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))) → reverseA_out_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421)

(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:

REVACCB_IN_GAGG(.(T472, T473), T477, T475, T476) → REVACCB_IN_GAGG(T473, T477, T472, .(T475, T476))

The TRS R consists of the following rules:

reverseA_in_ga([], []) → reverseA_out_ga([], [])
reverseA_in_ga(.(T36, []), .(T36, [])) → reverseA_out_ga(.(T36, []), .(T36, []))
reverseA_in_ga(.(T70, .(T69, [])), .(T69, .(T70, []))) → reverseA_out_ga(.(T70, .(T69, [])), .(T69, .(T70, [])))
reverseA_in_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))) → reverseA_out_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, []))))
reverseA_in_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))) → reverseA_out_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, [])))))
reverseA_in_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))) → reverseA_out_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, []))))))
reverseA_in_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))) → reverseA_out_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))))
reverseA_in_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))) → reverseA_out_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))))
reverseA_in_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U1_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
revaccB_in_gagg([], .(T460, T461), T460, T461) → revaccB_out_gagg([], .(T460, T461), T460, T461)
revaccB_in_gagg(.(T472, T473), T477, T475, T476) → U2_gagg(T472, T473, T477, T475, T476, revaccB_in_gagg(T473, T477, T472, .(T475, T476)))
U2_gagg(T472, T473, T477, T475, T476, revaccB_out_gagg(T473, T477, T472, .(T475, T476))) → revaccB_out_gagg(.(T472, T473), T477, T475, T476)
U1_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revaccB_out_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))) → reverseA_out_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421)

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)
revaccB_in_gagg(x1, x2, x3, x4) = revaccB_in_gagg(x1, x3, x4)
revaccB_out_gagg(x1, x2, x3, x4) = revaccB_out_gagg(x1, x2, x3, x4)
U2_gagg(x1, x2, x3, x4, x5, x6) = U2_gagg(x1, x2, x4, x5, x6)
REVACCB_IN_GAGG(x1, x2, x3, x4) = REVACCB_IN_GAGG(x1, x3, x4)

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:

REVACCB_IN_GAGG(.(T472, T473), T477, T475, T476) → REVACCB_IN_GAGG(T473, T477, T472, .(T475, T476))

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

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:

REVACCB_IN_GAGG(.(T472, T473), T475, T476) → REVACCB_IN_GAGG(T473, T472, .(T475, T476))

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:

REVACCB_IN_GAGG(.(T472, T473), T475, T476) → REVACCB_IN_GAGG(T473, T472, .(T475, T476))
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