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

(0) Obligation:

Clauses:

slowsort(X, Y) :- ','(perm(X, Y), sorted(Y)).
sorted([]).
sorted(.(X, [])).
sorted(.(X, .(Y, Z))) :- ','(le(X, Y), sorted(.(Y, Z))).
perm([], []).
perm(.(X, .(Y, [])), .(U, .(V, []))) :- ','(delete(U, .(X, Y), Z), perm(Z, V)).
delete(X, .(X, Y), Y).
delete(X, .(Y, Z), W) :- delete(X, Z, W).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(X)).
le(0, 0).

Query: slowsort(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: slowsort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: slowsort(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(perm(T5, T7), sorted(T7))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(perm(T5, T7), sorted(T7)), SCOPE: 2, CLAUSE: 4 | ','(perm(T5, T7), sorted(T7)), SCOPE: 2, CLAUSE: 5\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(perm(T5, T7), sorted(T7)), SCOPE: 2, CLAUSE: 4\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(perm(T5, T7), sorted(T7)), SCOPE: 2, CLAUSE: 5\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: sorted([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: sorted([]), SCOPE: 3, CLAUSE: 1 | sorted([]), SCOPE: 3, CLAUSE: 2 | sorted([]), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: sorted([]), SCOPE: 3, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: sorted([]), SCOPE: 3, CLAUSE: 2 | sorted([]), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: sorted([]), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="B: ','(','(delete(T20, .(T16, T17), X22), perm(X22, T21)), sorted(.(T20, .(T21, []))))\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="D: delete(T20, .(T16, T17), X22)\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="K: ','(perm(T26, T27), sorted(.(T28, .(T27, []))))\n\nT28 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: delete(T20, .(T16, T17), X22), SCOPE: 4, CLAUSE: 6 | delete(T20, .(T16, T17), X22), SCOPE: 4, CLAUSE: 7\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: delete(T20, .(T16, T17), X22), SCOPE: 4, CLAUSE: 6\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: delete(T20, .(T16, T17), X22), SCOPE: 4, CLAUSE: 7\n\nT16 is ground\nT17 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="C: delete(T57, T56, X59)\n\nT56 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: delete(T57, T56, X59), SCOPE: 5, CLAUSE: 6 | delete(T57, T56, X59), SCOPE: 5, CLAUSE: 7\n\nT56 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: delete(T57, T56, X59), SCOPE: 5, CLAUSE: 6\n\nT56 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: delete(T57, T56, X59), SCOPE: 5, CLAUSE: 7\n\nT56 is ground\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: delete(T81, T80, X88)\n\nT80 is ground\nX88 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="E: perm(T26, T27)\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="H: sorted(.(T28, .(T86, [])))\n\nT28 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: perm(T26, T27), SCOPE: 6, CLAUSE: 4 | perm(T26, T27), SCOPE: 6, CLAUSE: 5\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: perm(T26, T27), SCOPE: 6, CLAUSE: 4\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: perm(T26, T27), SCOPE: 6, CLAUSE: 5\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="F: ','(delete(T99, .(T95, T96), X106), perm(X106, T100))\n\nT95 is ground\nT96 is ground\nX106 is free", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: delete(T99, .(T95, T96), X106)\n\nT95 is ground\nT96 is ground\nX106 is free", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: perm(T105, T106)\n\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 1 | sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 2 | sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 3\n\nT28 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 2 | sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 3\n\nT28 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: sorted(.(T28, .(T86, []))), SCOPE: 7, CLAUSE: 3\n\nT28 is ground\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="I: ','(le(T118, T119), sorted(.(T119, [])))\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="G: le(T118, T119)\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="J: sorted(.(T119, []))\n\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="167: le(T118, T119), SCOPE: 8, CLAUSE: 8 | le(T118, T119), SCOPE: 8, CLAUSE: 9 | le(T118, T119), SCOPE: 8, CLAUSE: 10\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: le(T118, T119), SCOPE: 8, CLAUSE: 8\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: le(T118, T119), SCOPE: 8, CLAUSE: 9 | le(T118, T119), SCOPE: 8, CLAUSE: 10\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: le(T132, T133)\n\nT132 is ground\nT133 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
179 [label="179: le(T118, T119), SCOPE: 8, CLAUSE: 9\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
180 [label="180: le(T118, T119), SCOPE: 8, CLAUSE: 10\n\nT118 is ground\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
182 [label="182: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="183: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
184 [label="184: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
185 [label="185: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
186 [label="186: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
187 [label="187: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
190 [label="190: sorted(.(T119, [])), SCOPE: 9, CLAUSE: 1 | sorted(.(T119, [])), SCOPE: 9, CLAUSE: 2 | sorted(.(T119, [])), SCOPE: 9, CLAUSE: 3\n\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: sorted(.(T119, [])), SCOPE: 9, CLAUSE: 2 | sorted(.(T119, [])), SCOPE: 9, CLAUSE: 3\n\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
196 [label="196: sorted(.(T119, [])), SCOPE: 9, CLAUSE: 2\n\nT119 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="197: sorted(.(T119, [])), SCOPE: 9, CLAUSE: 3\n\nT119 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];
203 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 203 [arrowhead = none ];
203 -> 5;

204 [label="ONLY EVAL with clause\nslowsort(X3, X4) :- ','(perm(X3, X4), sorted(X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 204 [arrowhead = none ];
204 -> 8;

205 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 205 [arrowhead = none ];
205 -> 15;

206 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 206 [arrowhead = none ];
206 -> 16;

207 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 207 [arrowhead = none ];
207 -> 17;

208 [label="EVAL with clause\nperm([], []).\nand substitutionT5 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 208 [arrowhead = none ];
208 -> 18;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 209 [arrowhead = none ];
209 -> 19;

210 [label="EVAL with clause\nperm(.(X18, .(X19, [])), .(X20, .(X21, []))) :- ','(delete(X20, .(X18, X19), X22), perm(X22, X21)).\nand substitutionX18 -> T16,\nX19 -> T17,\nT5 -> .(T16, .(T17, [])),\nX20 -> T20,\nX21 -> T21,\nT7 -> .(T20, .(T21, [])),\nT18 -> T20,\nT19 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 210 [arrowhead = none ];
210 -> 42;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 211 [arrowhead = none ];
211 -> 43;

212 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 212 [arrowhead = none ];
212 -> 20;

213 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 213 [arrowhead = none ];
213 -> 21;

214 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 214 [arrowhead = none ];
214 -> 22;

215 [label="ONLY EVAL with clause\nsorted([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 215 [arrowhead = none ];
215 -> 23;

216 [label="BACKTRACK\nfor clause: sorted(.(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 216 [arrowhead = none ];
216 -> 27;

217 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 217 [arrowhead = none ];
217 -> 24;

218 [label="BACKTRACK\nfor clause: sorted(.(X, .(Y, Z))) :- ','(le(X, Y), sorted(.(Y, Z)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 218 [arrowhead = none ];
218 -> 28;

219 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
42 -> 219 [arrowhead = none ];
219 -> 46;

220 [label="SPLIT 2\nnew knowledge:\nT28 is ground\nT16 is ground\nT17 is ground\nT26 is ground\nreplacements:X22 -> T26,\nT21 -> T27,\nT20 -> T28", fontsize=14, style = filled, fillcolor = violet];
42 -> 220 [arrowhead = none ];
220 -> 48;

221 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 221 [arrowhead = none ];
221 -> 50;

222 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
48 -> 222 [arrowhead = none ];
222 -> 97;

223 [label="SPLIT 2\nnew knowledge:\nT26 is ground\nT86 is ground\nreplacements:T27 -> T86", fontsize=14, style = filled, fillcolor = violet];
48 -> 223 [arrowhead = none ];
223 -> 98;

224 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 224 [arrowhead = none ];
224 -> 51;

225 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 225 [arrowhead = none ];
225 -> 52;

226 [label="EVAL with clause\ndelete(X39, .(X39, X40), X40).\nand substitutionT20 -> T41,\nX39 -> T41,\nT16 -> T41,\nT17 -> T42,\nX40 -> T42,\nX22 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 226 [arrowhead = none ];
226 -> 63;

227 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 227 [arrowhead = none ];
227 -> 64;

228 [label="ONLY EVAL with clause\ndelete(X55, .(X56, X57), X58) :- delete(X55, X57, X58).\nand substitutionT20 -> T57,\nX55 -> T57,\nT16 -> T55,\nX56 -> T55,\nT17 -> T56,\nX57 -> T56,\nX22 -> X59,\nX58 -> X59,\nT54 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 228 [arrowhead = none ];
228 -> 67;

229 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 229 [arrowhead = none ];
229 -> 65;

230 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 230 [arrowhead = none ];
230 -> 68;

231 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 231 [arrowhead = none ];
231 -> 71;

232 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 232 [arrowhead = none ];
232 -> 72;

233 [label="EVAL with clause\ndelete(X72, .(X72, X73), X73).\nand substitutionT57 -> T70,\nX72 -> T70,\nX73 -> T71,\nT56 -> .(T70, T71),\nX59 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 233 [arrowhead = none ];
233 -> 76;

234 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 234 [arrowhead = none ];
234 -> 77;

235 [label="EVAL with clause\ndelete(X84, .(X85, X86), X87) :- delete(X84, X86, X87).\nand substitutionT57 -> T81,\nX84 -> T81,\nX85 -> T79,\nX86 -> T80,\nT56 -> .(T79, T80),\nX59 -> X88,\nX87 -> X88,\nT78 -> T81", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 235 [arrowhead = none ];
235 -> 80;

236 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 236 [arrowhead = none ];
236 -> 81;

237 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 237 [arrowhead = none ];
237 -> 78;

238 [label="INSTANCE with matching:\nT57 -> T81\nT56 -> T80\nX59 -> X88", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 238 [arrowhead = none , style=dashed];
238 -> 67[style=dashed];

239 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
97 -> 239 [arrowhead = none ];
239 -> 117;

240 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
98 -> 240 [arrowhead = none ];
240 -> 140;

241 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
117 -> 241 [arrowhead = none ];
241 -> 118;

242 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
117 -> 242 [arrowhead = none ];
242 -> 119;

243 [label="EVAL with clause\nperm([], []).\nand substitutionT26 -> [],\nT27 -> []", fontsize=14, style = filled, fillcolor = lightblue];
118 -> 243 [arrowhead = none ];
243 -> 120;

244 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
118 -> 244 [arrowhead = none ];
244 -> 121;

245 [label="EVAL with clause\nperm(.(X102, .(X103, [])), .(X104, .(X105, []))) :- ','(delete(X104, .(X102, X103), X106), perm(X106, X105)).\nand substitutionX102 -> T95,\nX103 -> T96,\nT26 -> .(T95, .(T96, [])),\nX104 -> T99,\nX105 -> T100,\nT27 -> .(T99, .(T100, [])),\nT97 -> T99,\nT98 -> T100", fontsize=14, style = filled, fillcolor = lightblue];
119 -> 245 [arrowhead = none ];
245 -> 125;

246 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
119 -> 246 [arrowhead = none ];
246 -> 126;

247 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
120 -> 247 [arrowhead = none ];
247 -> 122;

248 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
125 -> 248 [arrowhead = none ];
248 -> 127;

249 [label="SPLIT 2\nnew knowledge:\nT99 is ground\nT95 is ground\nT96 is ground\nT105 is ground\nreplacements:X106 -> T105,\nT100 -> T106", fontsize=14, style = filled, fillcolor = violet];
125 -> 249 [arrowhead = none ];
249 -> 128;

250 [label="INSTANCE with matching:\nT20 -> T99\nT16 -> T95\nT17 -> T96\nX22 -> X106", fontsize=14, style = filled, fillcolor = lightgrey];
127 -> 250 [arrowhead = none , style=dashed];
250 -> 46[style=dashed];

251 [label="INSTANCE with matching:\nT26 -> T105\nT27 -> T106", fontsize=14, style = filled, fillcolor = lightgrey];
128 -> 251 [arrowhead = none , style=dashed];
251 -> 97[style=dashed];

252 [label="BACKTRACK\nfor clause: sorted([])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
140 -> 252 [arrowhead = none ];
252 -> 141;

253 [label="BACKTRACK\nfor clause: sorted(.(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
141 -> 253 [arrowhead = none ];
253 -> 146;

254 [label="ONLY EVAL with clause\nsorted(.(X124, .(X125, X126))) :- ','(le(X124, X125), sorted(.(X125, X126))).\nand substitutionT28 -> T118,\nX124 -> T118,\nT86 -> T119,\nX125 -> T119,\nX126 -> []", fontsize=14, style = filled, fillcolor = lightblue];
146 -> 254 [arrowhead = none ];
254 -> 155;

255 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
155 -> 255 [arrowhead = none ];
255 -> 161;

256 [label="SPLIT 2\nnew knowledge:\nT118 is ground\nT119 is ground", fontsize=14, style = filled, fillcolor = violet];
155 -> 256 [arrowhead = none ];
256 -> 162;

257 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
161 -> 257 [arrowhead = none ];
257 -> 167;

258 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
162 -> 258 [arrowhead = none ];
258 -> 190;

259 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
167 -> 259 [arrowhead = none ];
259 -> 170;

260 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
167 -> 260 [arrowhead = none ];
260 -> 171;

261 [label="EVAL with clause\nle(s(X139), s(X140)) :- le(X139, X140).\nand substitutionX139 -> T132,\nT118 -> s(T132),\nX140 -> T133,\nT119 -> s(T133)", fontsize=14, style = filled, fillcolor = lightblue];
170 -> 261 [arrowhead = none ];
261 -> 175;

262 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
170 -> 262 [arrowhead = none ];
262 -> 177;

263 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
171 -> 263 [arrowhead = none ];
263 -> 179;

264 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
171 -> 264 [arrowhead = none ];
264 -> 180;

265 [label="INSTANCE with matching:\nT118 -> T132\nT119 -> T133", fontsize=14, style = filled, fillcolor = lightgrey];
175 -> 265 [arrowhead = none , style=dashed];
265 -> 161[style=dashed];

266 [label="EVAL with clause\nle(0, s(X147)).\nand substitutionT118 -> 0,\nX147 -> T140,\nT119 -> s(T140)", fontsize=14, style = filled, fillcolor = lightblue];
179 -> 266 [arrowhead = none ];
266 -> 182;

267 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
179 -> 267 [arrowhead = none ];
267 -> 183;

268 [label="EVAL with clause\nle(0, 0).\nand substitutionT118 -> 0,\nT119 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
180 -> 268 [arrowhead = none ];
268 -> 185;

269 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
180 -> 269 [arrowhead = none ];
269 -> 186;

270 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
182 -> 270 [arrowhead = none ];
270 -> 184;

271 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
185 -> 271 [arrowhead = none ];
271 -> 187;

272 [label="BACKTRACK\nfor clause: sorted([])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
190 -> 272 [arrowhead = none ];
272 -> 192;

273 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
192 -> 273 [arrowhead = none ];
273 -> 196;

274 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
192 -> 274 [arrowhead = none ];
274 -> 197;

275 [label="ONLY EVAL with clause\nsorted(.(X152, [])).\nand substitutionT119 -> T145,\nX152 -> T145", fontsize=14, style = filled, fillcolor = lightblue];
196 -> 275 [arrowhead = none ];
275 -> 200;

276 [label="BACKTRACK\nfor clause: sorted(.(X, .(Y, Z))) :- ','(le(X, Y), sorted(.(Y, Z)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
197 -> 276 [arrowhead = none ];
276 -> 202;

277 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
200 -> 277 [arrowhead = none ];
277 -> 201;

}

(2) Obligation:

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

slowsortA_in_ga([], []) → slowsortA_out_ga([], [])
slowsortA_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U1_ga(T16, T17, T20, T21, pB_in_aggaa(T20, T16, T17, X22, T21))
pB_in_aggaa(T28, T16, T17, T26, T27) → U7_aggaa(T28, T16, T17, T26, T27, deleteD_in_agga(T28, T16, T17, T26))
deleteD_in_agga(T41, T41, T42, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T57, T55, T56, X59) → U3_agga(T57, T55, T56, X59, deleteC_in_aga(T57, T56, X59))
deleteC_in_aga(T70, .(T70, T71), T71) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(T81, .(T79, T80), X88) → U2_aga(T81, T79, T80, X88, deleteC_in_aga(T81, T80, X88))
U2_aga(T81, T79, T80, X88, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)
U3_agga(T57, T55, T56, X59, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
U7_aggaa(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → U8_aggaa(T28, T16, T17, T26, T27, pK_in_gag(T26, T27, T28))
pK_in_gag(T26, T86, T28) → U9_gag(T26, T86, T28, permE_in_ga(T26, T86))
permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U4_ga(T95, T96, T99, T100, pF_in_aggaa(T99, T95, T96, X106, T100))
pF_in_aggaa(T99, T95, T96, T105, T106) → U11_aggaa(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
U11_aggaa(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → U12_aggaa(T99, T95, T96, T105, T106, permE_in_ga(T105, T106))
U12_aggaa(T99, T95, T96, T105, T106, permE_out_ga(T105, T106)) → pF_out_aggaa(T99, T95, T96, T105, T106)
U4_ga(T95, T96, T99, T100, pF_out_aggaa(T99, T95, T96, X106, T100)) → permE_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
U9_gag(T26, T86, T28, permE_out_ga(T26, T86)) → U10_gag(T26, T86, T28, sortedH_in_gg(T28, T86))
sortedH_in_gg(T118, T119) → U6_gg(T118, T119, pI_in_gg(T118, T119))
pI_in_gg(T118, T119) → U13_gg(T118, T119, leG_in_gg(T118, T119))
leG_in_gg(s(T132), s(T133)) → U5_gg(T132, T133, leG_in_gg(T132, T133))
leG_in_gg(0, s(T140)) → leG_out_gg(0, s(T140))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T132, T133, leG_out_gg(T132, T133)) → leG_out_gg(s(T132), s(T133))
U13_gg(T118, T119, leG_out_gg(T118, T119)) → U14_gg(T118, T119, sortedJ_in_g(T119))
sortedJ_in_g(T145) → sortedJ_out_g(T145)
U14_gg(T118, T119, sortedJ_out_g(T119)) → pI_out_gg(T118, T119)
U6_gg(T118, T119, pI_out_gg(T118, T119)) → sortedH_out_gg(T118, T119)
U10_gag(T26, T86, T28, sortedH_out_gg(T28, T86)) → pK_out_gag(T26, T86, T28)
U8_aggaa(T28, T16, T17, T26, T27, pK_out_gag(T26, T27, T28)) → pB_out_aggaa(T28, T16, T17, T26, T27)
U1_ga(T16, T17, T20, T21, pB_out_aggaa(T20, T16, T17, X22, T21)) → slowsortA_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))

The argument filtering Pi contains the following mapping:
slowsortA_in_ga(x1, x2)  =  slowsortA_in_ga(x1)
[]  =  []
slowsortA_out_ga(x1, x2)  =  slowsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_aggaa(x1, x2, x3, x4, x5)  =  pB_in_aggaa(x2, x3)
U7_aggaa(x1, x2, x3, x4, x5, x6)  =  U7_aggaa(x2, x3, x6)
deleteD_in_agga(x1, x2, x3, x4)  =  deleteD_in_agga(x2, x3)
deleteD_out_agga(x1, x2, x3, x4)  =  deleteD_out_agga(x1, x2, x3, x4)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x2, x3, x5)
U8_aggaa(x1, x2, x3, x4, x5, x6)  =  U8_aggaa(x1, x2, x3, x4, x6)
pK_in_gag(x1, x2, x3)  =  pK_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
permE_out_ga(x1, x2)  =  permE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
pF_in_aggaa(x1, x2, x3, x4, x5)  =  pF_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6)  =  U11_aggaa(x2, x3, x6)
U12_aggaa(x1, x2, x3, x4, x5, x6)  =  U12_aggaa(x1, x2, x3, x4, x6)
pF_out_aggaa(x1, x2, x3, x4, x5)  =  pF_out_aggaa(x1, x2, x3, x4, x5)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x2, x3, x4)
sortedH_in_gg(x1, x2)  =  sortedH_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
pI_in_gg(x1, x2)  =  pI_in_gg(x1, x2)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
leG_in_gg(x1, x2)  =  leG_in_gg(x1, x2)
s(x1)  =  s(x1)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
0  =  0
leG_out_gg(x1, x2)  =  leG_out_gg(x1, x2)
U14_gg(x1, x2, x3)  =  U14_gg(x1, x2, x3)
sortedJ_in_g(x1)  =  sortedJ_in_g(x1)
sortedJ_out_g(x1)  =  sortedJ_out_g(x1)
pI_out_gg(x1, x2)  =  pI_out_gg(x1, x2)
sortedH_out_gg(x1, x2)  =  sortedH_out_gg(x1, x2)
pK_out_gag(x1, x2, x3)  =  pK_out_gag(x1, x2, x3)
pB_out_aggaa(x1, x2, x3, x4, x5)  =  pB_out_aggaa(x1, x2, x3, x4, x5)

(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:

SLOWSORTA_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → U1_GA(T16, T17, T20, T21, pB_in_aggaa(T20, T16, T17, X22, T21))
SLOWSORTA_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → PB_IN_AGGAA(T20, T16, T17, X22, T21)
PB_IN_AGGAA(T28, T16, T17, T26, T27) → U7_AGGAA(T28, T16, T17, T26, T27, deleteD_in_agga(T28, T16, T17, T26))
PB_IN_AGGAA(T28, T16, T17, T26, T27) → DELETED_IN_AGGA(T28, T16, T17, T26)
DELETED_IN_AGGA(T57, T55, T56, X59) → U3_AGGA(T57, T55, T56, X59, deleteC_in_aga(T57, T56, X59))
DELETED_IN_AGGA(T57, T55, T56, X59) → DELETEC_IN_AGA(T57, T56, X59)
DELETEC_IN_AGA(T81, .(T79, T80), X88) → U2_AGA(T81, T79, T80, X88, deleteC_in_aga(T81, T80, X88))
DELETEC_IN_AGA(T81, .(T79, T80), X88) → DELETEC_IN_AGA(T81, T80, X88)
U7_AGGAA(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → U8_AGGAA(T28, T16, T17, T26, T27, pK_in_gag(T26, T27, T28))
U7_AGGAA(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → PK_IN_GAG(T26, T27, T28)
PK_IN_GAG(T26, T86, T28) → U9_GAG(T26, T86, T28, permE_in_ga(T26, T86))
PK_IN_GAG(T26, T86, T28) → PERME_IN_GA(T26, T86)
PERME_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → U4_GA(T95, T96, T99, T100, pF_in_aggaa(T99, T95, T96, X106, T100))
PERME_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → PF_IN_AGGAA(T99, T95, T96, X106, T100)
PF_IN_AGGAA(T99, T95, T96, T105, T106) → U11_AGGAA(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
PF_IN_AGGAA(T99, T95, T96, T105, T106) → DELETED_IN_AGGA(T99, T95, T96, T105)
U11_AGGAA(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → U12_AGGAA(T99, T95, T96, T105, T106, permE_in_ga(T105, T106))
U11_AGGAA(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → PERME_IN_GA(T105, T106)
U9_GAG(T26, T86, T28, permE_out_ga(T26, T86)) → U10_GAG(T26, T86, T28, sortedH_in_gg(T28, T86))
U9_GAG(T26, T86, T28, permE_out_ga(T26, T86)) → SORTEDH_IN_GG(T28, T86)
SORTEDH_IN_GG(T118, T119) → U6_GG(T118, T119, pI_in_gg(T118, T119))
SORTEDH_IN_GG(T118, T119) → PI_IN_GG(T118, T119)
PI_IN_GG(T118, T119) → U13_GG(T118, T119, leG_in_gg(T118, T119))
PI_IN_GG(T118, T119) → LEG_IN_GG(T118, T119)
LEG_IN_GG(s(T132), s(T133)) → U5_GG(T132, T133, leG_in_gg(T132, T133))
LEG_IN_GG(s(T132), s(T133)) → LEG_IN_GG(T132, T133)
U13_GG(T118, T119, leG_out_gg(T118, T119)) → U14_GG(T118, T119, sortedJ_in_g(T119))
U13_GG(T118, T119, leG_out_gg(T118, T119)) → SORTEDJ_IN_G(T119)

The TRS R consists of the following rules:

slowsortA_in_ga([], []) → slowsortA_out_ga([], [])
slowsortA_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U1_ga(T16, T17, T20, T21, pB_in_aggaa(T20, T16, T17, X22, T21))
pB_in_aggaa(T28, T16, T17, T26, T27) → U7_aggaa(T28, T16, T17, T26, T27, deleteD_in_agga(T28, T16, T17, T26))
deleteD_in_agga(T41, T41, T42, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T57, T55, T56, X59) → U3_agga(T57, T55, T56, X59, deleteC_in_aga(T57, T56, X59))
deleteC_in_aga(T70, .(T70, T71), T71) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(T81, .(T79, T80), X88) → U2_aga(T81, T79, T80, X88, deleteC_in_aga(T81, T80, X88))
U2_aga(T81, T79, T80, X88, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)
U3_agga(T57, T55, T56, X59, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
U7_aggaa(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → U8_aggaa(T28, T16, T17, T26, T27, pK_in_gag(T26, T27, T28))
pK_in_gag(T26, T86, T28) → U9_gag(T26, T86, T28, permE_in_ga(T26, T86))
permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U4_ga(T95, T96, T99, T100, pF_in_aggaa(T99, T95, T96, X106, T100))
pF_in_aggaa(T99, T95, T96, T105, T106) → U11_aggaa(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
U11_aggaa(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → U12_aggaa(T99, T95, T96, T105, T106, permE_in_ga(T105, T106))
U12_aggaa(T99, T95, T96, T105, T106, permE_out_ga(T105, T106)) → pF_out_aggaa(T99, T95, T96, T105, T106)
U4_ga(T95, T96, T99, T100, pF_out_aggaa(T99, T95, T96, X106, T100)) → permE_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
U9_gag(T26, T86, T28, permE_out_ga(T26, T86)) → U10_gag(T26, T86, T28, sortedH_in_gg(T28, T86))
sortedH_in_gg(T118, T119) → U6_gg(T118, T119, pI_in_gg(T118, T119))
pI_in_gg(T118, T119) → U13_gg(T118, T119, leG_in_gg(T118, T119))
leG_in_gg(s(T132), s(T133)) → U5_gg(T132, T133, leG_in_gg(T132, T133))
leG_in_gg(0, s(T140)) → leG_out_gg(0, s(T140))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T132, T133, leG_out_gg(T132, T133)) → leG_out_gg(s(T132), s(T133))
U13_gg(T118, T119, leG_out_gg(T118, T119)) → U14_gg(T118, T119, sortedJ_in_g(T119))
sortedJ_in_g(T145) → sortedJ_out_g(T145)
U14_gg(T118, T119, sortedJ_out_g(T119)) → pI_out_gg(T118, T119)
U6_gg(T118, T119, pI_out_gg(T118, T119)) → sortedH_out_gg(T118, T119)
U10_gag(T26, T86, T28, sortedH_out_gg(T28, T86)) → pK_out_gag(T26, T86, T28)
U8_aggaa(T28, T16, T17, T26, T27, pK_out_gag(T26, T27, T28)) → pB_out_aggaa(T28, T16, T17, T26, T27)
U1_ga(T16, T17, T20, T21, pB_out_aggaa(T20, T16, T17, X22, T21)) → slowsortA_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))

The argument filtering Pi contains the following mapping:
slowsortA_in_ga(x1, x2)  =  slowsortA_in_ga(x1)
[]  =  []
slowsortA_out_ga(x1, x2)  =  slowsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_aggaa(x1, x2, x3, x4, x5)  =  pB_in_aggaa(x2, x3)
U7_aggaa(x1, x2, x3, x4, x5, x6)  =  U7_aggaa(x2, x3, x6)
deleteD_in_agga(x1, x2, x3, x4)  =  deleteD_in_agga(x2, x3)
deleteD_out_agga(x1, x2, x3, x4)  =  deleteD_out_agga(x1, x2, x3, x4)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x2, x3, x5)
U8_aggaa(x1, x2, x3, x4, x5, x6)  =  U8_aggaa(x1, x2, x3, x4, x6)
pK_in_gag(x1, x2, x3)  =  pK_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
permE_out_ga(x1, x2)  =  permE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
pF_in_aggaa(x1, x2, x3, x4, x5)  =  pF_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6)  =  U11_aggaa(x2, x3, x6)
U12_aggaa(x1, x2, x3, x4, x5, x6)  =  U12_aggaa(x1, x2, x3, x4, x6)
pF_out_aggaa(x1, x2, x3, x4, x5)  =  pF_out_aggaa(x1, x2, x3, x4, x5)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x2, x3, x4)
sortedH_in_gg(x1, x2)  =  sortedH_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
pI_in_gg(x1, x2)  =  pI_in_gg(x1, x2)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
leG_in_gg(x1, x2)  =  leG_in_gg(x1, x2)
s(x1)  =  s(x1)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
0  =  0
leG_out_gg(x1, x2)  =  leG_out_gg(x1, x2)
U14_gg(x1, x2, x3)  =  U14_gg(x1, x2, x3)
sortedJ_in_g(x1)  =  sortedJ_in_g(x1)
sortedJ_out_g(x1)  =  sortedJ_out_g(x1)
pI_out_gg(x1, x2)  =  pI_out_gg(x1, x2)
sortedH_out_gg(x1, x2)  =  sortedH_out_gg(x1, x2)
pK_out_gag(x1, x2, x3)  =  pK_out_gag(x1, x2, x3)
pB_out_aggaa(x1, x2, x3, x4, x5)  =  pB_out_aggaa(x1, x2, x3, x4, x5)
SLOWSORTA_IN_GA(x1, x2)  =  SLOWSORTA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
PB_IN_AGGAA(x1, x2, x3, x4, x5)  =  PB_IN_AGGAA(x2, x3)
U7_AGGAA(x1, x2, x3, x4, x5, x6)  =  U7_AGGAA(x2, x3, x6)
DELETED_IN_AGGA(x1, x2, x3, x4)  =  DELETED_IN_AGGA(x2, x3)
U3_AGGA(x1, x2, x3, x4, x5)  =  U3_AGGA(x2, x3, x5)
DELETEC_IN_AGA(x1, x2, x3)  =  DELETEC_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x2, x3, x5)
U8_AGGAA(x1, x2, x3, x4, x5, x6)  =  U8_AGGAA(x1, x2, x3, x4, x6)
PK_IN_GAG(x1, x2, x3)  =  PK_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x1, x3, x4)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
PF_IN_AGGAA(x1, x2, x3, x4, x5)  =  PF_IN_AGGAA(x2, x3)
U11_AGGAA(x1, x2, x3, x4, x5, x6)  =  U11_AGGAA(x2, x3, x6)
U12_AGGAA(x1, x2, x3, x4, x5, x6)  =  U12_AGGAA(x1, x2, x3, x4, x6)
U10_GAG(x1, x2, x3, x4)  =  U10_GAG(x1, x2, x3, x4)
SORTEDH_IN_GG(x1, x2)  =  SORTEDH_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
PI_IN_GG(x1, x2)  =  PI_IN_GG(x1, x2)
U13_GG(x1, x2, x3)  =  U13_GG(x1, x2, x3)
LEG_IN_GG(x1, x2)  =  LEG_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U14_GG(x1, x2, x3)  =  U14_GG(x1, x2, x3)
SORTEDJ_IN_G(x1)  =  SORTEDJ_IN_G(x1)

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

(4) Obligation:

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

SLOWSORTA_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → U1_GA(T16, T17, T20, T21, pB_in_aggaa(T20, T16, T17, X22, T21))
SLOWSORTA_IN_GA(.(T16, .(T17, [])), .(T20, .(T21, []))) → PB_IN_AGGAA(T20, T16, T17, X22, T21)
PB_IN_AGGAA(T28, T16, T17, T26, T27) → U7_AGGAA(T28, T16, T17, T26, T27, deleteD_in_agga(T28, T16, T17, T26))
PB_IN_AGGAA(T28, T16, T17, T26, T27) → DELETED_IN_AGGA(T28, T16, T17, T26)
DELETED_IN_AGGA(T57, T55, T56, X59) → U3_AGGA(T57, T55, T56, X59, deleteC_in_aga(T57, T56, X59))
DELETED_IN_AGGA(T57, T55, T56, X59) → DELETEC_IN_AGA(T57, T56, X59)
DELETEC_IN_AGA(T81, .(T79, T80), X88) → U2_AGA(T81, T79, T80, X88, deleteC_in_aga(T81, T80, X88))
DELETEC_IN_AGA(T81, .(T79, T80), X88) → DELETEC_IN_AGA(T81, T80, X88)
U7_AGGAA(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → U8_AGGAA(T28, T16, T17, T26, T27, pK_in_gag(T26, T27, T28))
U7_AGGAA(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → PK_IN_GAG(T26, T27, T28)
PK_IN_GAG(T26, T86, T28) → U9_GAG(T26, T86, T28, permE_in_ga(T26, T86))
PK_IN_GAG(T26, T86, T28) → PERME_IN_GA(T26, T86)
PERME_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → U4_GA(T95, T96, T99, T100, pF_in_aggaa(T99, T95, T96, X106, T100))
PERME_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → PF_IN_AGGAA(T99, T95, T96, X106, T100)
PF_IN_AGGAA(T99, T95, T96, T105, T106) → U11_AGGAA(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
PF_IN_AGGAA(T99, T95, T96, T105, T106) → DELETED_IN_AGGA(T99, T95, T96, T105)
U11_AGGAA(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → U12_AGGAA(T99, T95, T96, T105, T106, permE_in_ga(T105, T106))
U11_AGGAA(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → PERME_IN_GA(T105, T106)
U9_GAG(T26, T86, T28, permE_out_ga(T26, T86)) → U10_GAG(T26, T86, T28, sortedH_in_gg(T28, T86))
U9_GAG(T26, T86, T28, permE_out_ga(T26, T86)) → SORTEDH_IN_GG(T28, T86)
SORTEDH_IN_GG(T118, T119) → U6_GG(T118, T119, pI_in_gg(T118, T119))
SORTEDH_IN_GG(T118, T119) → PI_IN_GG(T118, T119)
PI_IN_GG(T118, T119) → U13_GG(T118, T119, leG_in_gg(T118, T119))
PI_IN_GG(T118, T119) → LEG_IN_GG(T118, T119)
LEG_IN_GG(s(T132), s(T133)) → U5_GG(T132, T133, leG_in_gg(T132, T133))
LEG_IN_GG(s(T132), s(T133)) → LEG_IN_GG(T132, T133)
U13_GG(T118, T119, leG_out_gg(T118, T119)) → U14_GG(T118, T119, sortedJ_in_g(T119))
U13_GG(T118, T119, leG_out_gg(T118, T119)) → SORTEDJ_IN_G(T119)

The TRS R consists of the following rules:

slowsortA_in_ga([], []) → slowsortA_out_ga([], [])
slowsortA_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U1_ga(T16, T17, T20, T21, pB_in_aggaa(T20, T16, T17, X22, T21))
pB_in_aggaa(T28, T16, T17, T26, T27) → U7_aggaa(T28, T16, T17, T26, T27, deleteD_in_agga(T28, T16, T17, T26))
deleteD_in_agga(T41, T41, T42, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T57, T55, T56, X59) → U3_agga(T57, T55, T56, X59, deleteC_in_aga(T57, T56, X59))
deleteC_in_aga(T70, .(T70, T71), T71) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(T81, .(T79, T80), X88) → U2_aga(T81, T79, T80, X88, deleteC_in_aga(T81, T80, X88))
U2_aga(T81, T79, T80, X88, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)
U3_agga(T57, T55, T56, X59, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
U7_aggaa(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → U8_aggaa(T28, T16, T17, T26, T27, pK_in_gag(T26, T27, T28))
pK_in_gag(T26, T86, T28) → U9_gag(T26, T86, T28, permE_in_ga(T26, T86))
permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U4_ga(T95, T96, T99, T100, pF_in_aggaa(T99, T95, T96, X106, T100))
pF_in_aggaa(T99, T95, T96, T105, T106) → U11_aggaa(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
U11_aggaa(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → U12_aggaa(T99, T95, T96, T105, T106, permE_in_ga(T105, T106))
U12_aggaa(T99, T95, T96, T105, T106, permE_out_ga(T105, T106)) → pF_out_aggaa(T99, T95, T96, T105, T106)
U4_ga(T95, T96, T99, T100, pF_out_aggaa(T99, T95, T96, X106, T100)) → permE_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
U9_gag(T26, T86, T28, permE_out_ga(T26, T86)) → U10_gag(T26, T86, T28, sortedH_in_gg(T28, T86))
sortedH_in_gg(T118, T119) → U6_gg(T118, T119, pI_in_gg(T118, T119))
pI_in_gg(T118, T119) → U13_gg(T118, T119, leG_in_gg(T118, T119))
leG_in_gg(s(T132), s(T133)) → U5_gg(T132, T133, leG_in_gg(T132, T133))
leG_in_gg(0, s(T140)) → leG_out_gg(0, s(T140))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T132, T133, leG_out_gg(T132, T133)) → leG_out_gg(s(T132), s(T133))
U13_gg(T118, T119, leG_out_gg(T118, T119)) → U14_gg(T118, T119, sortedJ_in_g(T119))
sortedJ_in_g(T145) → sortedJ_out_g(T145)
U14_gg(T118, T119, sortedJ_out_g(T119)) → pI_out_gg(T118, T119)
U6_gg(T118, T119, pI_out_gg(T118, T119)) → sortedH_out_gg(T118, T119)
U10_gag(T26, T86, T28, sortedH_out_gg(T28, T86)) → pK_out_gag(T26, T86, T28)
U8_aggaa(T28, T16, T17, T26, T27, pK_out_gag(T26, T27, T28)) → pB_out_aggaa(T28, T16, T17, T26, T27)
U1_ga(T16, T17, T20, T21, pB_out_aggaa(T20, T16, T17, X22, T21)) → slowsortA_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))

The argument filtering Pi contains the following mapping:
slowsortA_in_ga(x1, x2)  =  slowsortA_in_ga(x1)
[]  =  []
slowsortA_out_ga(x1, x2)  =  slowsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_aggaa(x1, x2, x3, x4, x5)  =  pB_in_aggaa(x2, x3)
U7_aggaa(x1, x2, x3, x4, x5, x6)  =  U7_aggaa(x2, x3, x6)
deleteD_in_agga(x1, x2, x3, x4)  =  deleteD_in_agga(x2, x3)
deleteD_out_agga(x1, x2, x3, x4)  =  deleteD_out_agga(x1, x2, x3, x4)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x2, x3, x5)
U8_aggaa(x1, x2, x3, x4, x5, x6)  =  U8_aggaa(x1, x2, x3, x4, x6)
pK_in_gag(x1, x2, x3)  =  pK_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
permE_out_ga(x1, x2)  =  permE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
pF_in_aggaa(x1, x2, x3, x4, x5)  =  pF_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6)  =  U11_aggaa(x2, x3, x6)
U12_aggaa(x1, x2, x3, x4, x5, x6)  =  U12_aggaa(x1, x2, x3, x4, x6)
pF_out_aggaa(x1, x2, x3, x4, x5)  =  pF_out_aggaa(x1, x2, x3, x4, x5)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x2, x3, x4)
sortedH_in_gg(x1, x2)  =  sortedH_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
pI_in_gg(x1, x2)  =  pI_in_gg(x1, x2)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
leG_in_gg(x1, x2)  =  leG_in_gg(x1, x2)
s(x1)  =  s(x1)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
0  =  0
leG_out_gg(x1, x2)  =  leG_out_gg(x1, x2)
U14_gg(x1, x2, x3)  =  U14_gg(x1, x2, x3)
sortedJ_in_g(x1)  =  sortedJ_in_g(x1)
sortedJ_out_g(x1)  =  sortedJ_out_g(x1)
pI_out_gg(x1, x2)  =  pI_out_gg(x1, x2)
sortedH_out_gg(x1, x2)  =  sortedH_out_gg(x1, x2)
pK_out_gag(x1, x2, x3)  =  pK_out_gag(x1, x2, x3)
pB_out_aggaa(x1, x2, x3, x4, x5)  =  pB_out_aggaa(x1, x2, x3, x4, x5)
SLOWSORTA_IN_GA(x1, x2)  =  SLOWSORTA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
PB_IN_AGGAA(x1, x2, x3, x4, x5)  =  PB_IN_AGGAA(x2, x3)
U7_AGGAA(x1, x2, x3, x4, x5, x6)  =  U7_AGGAA(x2, x3, x6)
DELETED_IN_AGGA(x1, x2, x3, x4)  =  DELETED_IN_AGGA(x2, x3)
U3_AGGA(x1, x2, x3, x4, x5)  =  U3_AGGA(x2, x3, x5)
DELETEC_IN_AGA(x1, x2, x3)  =  DELETEC_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x2, x3, x5)
U8_AGGAA(x1, x2, x3, x4, x5, x6)  =  U8_AGGAA(x1, x2, x3, x4, x6)
PK_IN_GAG(x1, x2, x3)  =  PK_IN_GAG(x1, x3)
U9_GAG(x1, x2, x3, x4)  =  U9_GAG(x1, x3, x4)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
PF_IN_AGGAA(x1, x2, x3, x4, x5)  =  PF_IN_AGGAA(x2, x3)
U11_AGGAA(x1, x2, x3, x4, x5, x6)  =  U11_AGGAA(x2, x3, x6)
U12_AGGAA(x1, x2, x3, x4, x5, x6)  =  U12_AGGAA(x1, x2, x3, x4, x6)
U10_GAG(x1, x2, x3, x4)  =  U10_GAG(x1, x2, x3, x4)
SORTEDH_IN_GG(x1, x2)  =  SORTEDH_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
PI_IN_GG(x1, x2)  =  PI_IN_GG(x1, x2)
U13_GG(x1, x2, x3)  =  U13_GG(x1, x2, x3)
LEG_IN_GG(x1, x2)  =  LEG_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U14_GG(x1, x2, x3)  =  U14_GG(x1, x2, x3)
SORTEDJ_IN_G(x1)  =  SORTEDJ_IN_G(x1)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 23 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LEG_IN_GG(s(T132), s(T133)) → LEG_IN_GG(T132, T133)

The TRS R consists of the following rules:

slowsortA_in_ga([], []) → slowsortA_out_ga([], [])
slowsortA_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U1_ga(T16, T17, T20, T21, pB_in_aggaa(T20, T16, T17, X22, T21))
pB_in_aggaa(T28, T16, T17, T26, T27) → U7_aggaa(T28, T16, T17, T26, T27, deleteD_in_agga(T28, T16, T17, T26))
deleteD_in_agga(T41, T41, T42, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T57, T55, T56, X59) → U3_agga(T57, T55, T56, X59, deleteC_in_aga(T57, T56, X59))
deleteC_in_aga(T70, .(T70, T71), T71) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(T81, .(T79, T80), X88) → U2_aga(T81, T79, T80, X88, deleteC_in_aga(T81, T80, X88))
U2_aga(T81, T79, T80, X88, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)
U3_agga(T57, T55, T56, X59, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
U7_aggaa(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → U8_aggaa(T28, T16, T17, T26, T27, pK_in_gag(T26, T27, T28))
pK_in_gag(T26, T86, T28) → U9_gag(T26, T86, T28, permE_in_ga(T26, T86))
permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U4_ga(T95, T96, T99, T100, pF_in_aggaa(T99, T95, T96, X106, T100))
pF_in_aggaa(T99, T95, T96, T105, T106) → U11_aggaa(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
U11_aggaa(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → U12_aggaa(T99, T95, T96, T105, T106, permE_in_ga(T105, T106))
U12_aggaa(T99, T95, T96, T105, T106, permE_out_ga(T105, T106)) → pF_out_aggaa(T99, T95, T96, T105, T106)
U4_ga(T95, T96, T99, T100, pF_out_aggaa(T99, T95, T96, X106, T100)) → permE_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
U9_gag(T26, T86, T28, permE_out_ga(T26, T86)) → U10_gag(T26, T86, T28, sortedH_in_gg(T28, T86))
sortedH_in_gg(T118, T119) → U6_gg(T118, T119, pI_in_gg(T118, T119))
pI_in_gg(T118, T119) → U13_gg(T118, T119, leG_in_gg(T118, T119))
leG_in_gg(s(T132), s(T133)) → U5_gg(T132, T133, leG_in_gg(T132, T133))
leG_in_gg(0, s(T140)) → leG_out_gg(0, s(T140))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T132, T133, leG_out_gg(T132, T133)) → leG_out_gg(s(T132), s(T133))
U13_gg(T118, T119, leG_out_gg(T118, T119)) → U14_gg(T118, T119, sortedJ_in_g(T119))
sortedJ_in_g(T145) → sortedJ_out_g(T145)
U14_gg(T118, T119, sortedJ_out_g(T119)) → pI_out_gg(T118, T119)
U6_gg(T118, T119, pI_out_gg(T118, T119)) → sortedH_out_gg(T118, T119)
U10_gag(T26, T86, T28, sortedH_out_gg(T28, T86)) → pK_out_gag(T26, T86, T28)
U8_aggaa(T28, T16, T17, T26, T27, pK_out_gag(T26, T27, T28)) → pB_out_aggaa(T28, T16, T17, T26, T27)
U1_ga(T16, T17, T20, T21, pB_out_aggaa(T20, T16, T17, X22, T21)) → slowsortA_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))

The argument filtering Pi contains the following mapping:
slowsortA_in_ga(x1, x2)  =  slowsortA_in_ga(x1)
[]  =  []
slowsortA_out_ga(x1, x2)  =  slowsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_aggaa(x1, x2, x3, x4, x5)  =  pB_in_aggaa(x2, x3)
U7_aggaa(x1, x2, x3, x4, x5, x6)  =  U7_aggaa(x2, x3, x6)
deleteD_in_agga(x1, x2, x3, x4)  =  deleteD_in_agga(x2, x3)
deleteD_out_agga(x1, x2, x3, x4)  =  deleteD_out_agga(x1, x2, x3, x4)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x2, x3, x5)
U8_aggaa(x1, x2, x3, x4, x5, x6)  =  U8_aggaa(x1, x2, x3, x4, x6)
pK_in_gag(x1, x2, x3)  =  pK_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
permE_out_ga(x1, x2)  =  permE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
pF_in_aggaa(x1, x2, x3, x4, x5)  =  pF_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6)  =  U11_aggaa(x2, x3, x6)
U12_aggaa(x1, x2, x3, x4, x5, x6)  =  U12_aggaa(x1, x2, x3, x4, x6)
pF_out_aggaa(x1, x2, x3, x4, x5)  =  pF_out_aggaa(x1, x2, x3, x4, x5)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x2, x3, x4)
sortedH_in_gg(x1, x2)  =  sortedH_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
pI_in_gg(x1, x2)  =  pI_in_gg(x1, x2)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
leG_in_gg(x1, x2)  =  leG_in_gg(x1, x2)
s(x1)  =  s(x1)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
0  =  0
leG_out_gg(x1, x2)  =  leG_out_gg(x1, x2)
U14_gg(x1, x2, x3)  =  U14_gg(x1, x2, x3)
sortedJ_in_g(x1)  =  sortedJ_in_g(x1)
sortedJ_out_g(x1)  =  sortedJ_out_g(x1)
pI_out_gg(x1, x2)  =  pI_out_gg(x1, x2)
sortedH_out_gg(x1, x2)  =  sortedH_out_gg(x1, x2)
pK_out_gag(x1, x2, x3)  =  pK_out_gag(x1, x2, x3)
pB_out_aggaa(x1, x2, x3, x4, x5)  =  pB_out_aggaa(x1, x2, x3, x4, x5)
LEG_IN_GG(x1, x2)  =  LEG_IN_GG(x1, x2)

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:

LEG_IN_GG(s(T132), s(T133)) → LEG_IN_GG(T132, T133)

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

(10) PiDPToQDPProof (EQUIVALENT 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:

LEG_IN_GG(s(T132), s(T133)) → LEG_IN_GG(T132, T133)

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:

• LEG_IN_GG(s(T132), s(T133)) → LEG_IN_GG(T132, T133)
  The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

DELETEC_IN_AGA(T81, .(T79, T80), X88) → DELETEC_IN_AGA(T81, T80, X88)

The TRS R consists of the following rules:

slowsortA_in_ga([], []) → slowsortA_out_ga([], [])
slowsortA_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U1_ga(T16, T17, T20, T21, pB_in_aggaa(T20, T16, T17, X22, T21))
pB_in_aggaa(T28, T16, T17, T26, T27) → U7_aggaa(T28, T16, T17, T26, T27, deleteD_in_agga(T28, T16, T17, T26))
deleteD_in_agga(T41, T41, T42, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T57, T55, T56, X59) → U3_agga(T57, T55, T56, X59, deleteC_in_aga(T57, T56, X59))
deleteC_in_aga(T70, .(T70, T71), T71) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(T81, .(T79, T80), X88) → U2_aga(T81, T79, T80, X88, deleteC_in_aga(T81, T80, X88))
U2_aga(T81, T79, T80, X88, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)
U3_agga(T57, T55, T56, X59, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
U7_aggaa(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → U8_aggaa(T28, T16, T17, T26, T27, pK_in_gag(T26, T27, T28))
pK_in_gag(T26, T86, T28) → U9_gag(T26, T86, T28, permE_in_ga(T26, T86))
permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U4_ga(T95, T96, T99, T100, pF_in_aggaa(T99, T95, T96, X106, T100))
pF_in_aggaa(T99, T95, T96, T105, T106) → U11_aggaa(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
U11_aggaa(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → U12_aggaa(T99, T95, T96, T105, T106, permE_in_ga(T105, T106))
U12_aggaa(T99, T95, T96, T105, T106, permE_out_ga(T105, T106)) → pF_out_aggaa(T99, T95, T96, T105, T106)
U4_ga(T95, T96, T99, T100, pF_out_aggaa(T99, T95, T96, X106, T100)) → permE_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
U9_gag(T26, T86, T28, permE_out_ga(T26, T86)) → U10_gag(T26, T86, T28, sortedH_in_gg(T28, T86))
sortedH_in_gg(T118, T119) → U6_gg(T118, T119, pI_in_gg(T118, T119))
pI_in_gg(T118, T119) → U13_gg(T118, T119, leG_in_gg(T118, T119))
leG_in_gg(s(T132), s(T133)) → U5_gg(T132, T133, leG_in_gg(T132, T133))
leG_in_gg(0, s(T140)) → leG_out_gg(0, s(T140))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T132, T133, leG_out_gg(T132, T133)) → leG_out_gg(s(T132), s(T133))
U13_gg(T118, T119, leG_out_gg(T118, T119)) → U14_gg(T118, T119, sortedJ_in_g(T119))
sortedJ_in_g(T145) → sortedJ_out_g(T145)
U14_gg(T118, T119, sortedJ_out_g(T119)) → pI_out_gg(T118, T119)
U6_gg(T118, T119, pI_out_gg(T118, T119)) → sortedH_out_gg(T118, T119)
U10_gag(T26, T86, T28, sortedH_out_gg(T28, T86)) → pK_out_gag(T26, T86, T28)
U8_aggaa(T28, T16, T17, T26, T27, pK_out_gag(T26, T27, T28)) → pB_out_aggaa(T28, T16, T17, T26, T27)
U1_ga(T16, T17, T20, T21, pB_out_aggaa(T20, T16, T17, X22, T21)) → slowsortA_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))

The argument filtering Pi contains the following mapping:
slowsortA_in_ga(x1, x2)  =  slowsortA_in_ga(x1)
[]  =  []
slowsortA_out_ga(x1, x2)  =  slowsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_aggaa(x1, x2, x3, x4, x5)  =  pB_in_aggaa(x2, x3)
U7_aggaa(x1, x2, x3, x4, x5, x6)  =  U7_aggaa(x2, x3, x6)
deleteD_in_agga(x1, x2, x3, x4)  =  deleteD_in_agga(x2, x3)
deleteD_out_agga(x1, x2, x3, x4)  =  deleteD_out_agga(x1, x2, x3, x4)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x2, x3, x5)
U8_aggaa(x1, x2, x3, x4, x5, x6)  =  U8_aggaa(x1, x2, x3, x4, x6)
pK_in_gag(x1, x2, x3)  =  pK_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
permE_out_ga(x1, x2)  =  permE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
pF_in_aggaa(x1, x2, x3, x4, x5)  =  pF_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6)  =  U11_aggaa(x2, x3, x6)
U12_aggaa(x1, x2, x3, x4, x5, x6)  =  U12_aggaa(x1, x2, x3, x4, x6)
pF_out_aggaa(x1, x2, x3, x4, x5)  =  pF_out_aggaa(x1, x2, x3, x4, x5)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x2, x3, x4)
sortedH_in_gg(x1, x2)  =  sortedH_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
pI_in_gg(x1, x2)  =  pI_in_gg(x1, x2)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
leG_in_gg(x1, x2)  =  leG_in_gg(x1, x2)
s(x1)  =  s(x1)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
0  =  0
leG_out_gg(x1, x2)  =  leG_out_gg(x1, x2)
U14_gg(x1, x2, x3)  =  U14_gg(x1, x2, x3)
sortedJ_in_g(x1)  =  sortedJ_in_g(x1)
sortedJ_out_g(x1)  =  sortedJ_out_g(x1)
pI_out_gg(x1, x2)  =  pI_out_gg(x1, x2)
sortedH_out_gg(x1, x2)  =  sortedH_out_gg(x1, x2)
pK_out_gag(x1, x2, x3)  =  pK_out_gag(x1, x2, x3)
pB_out_aggaa(x1, x2, x3, x4, x5)  =  pB_out_aggaa(x1, x2, x3, x4, x5)
DELETEC_IN_AGA(x1, x2, x3)  =  DELETEC_IN_AGA(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:

DELETEC_IN_AGA(T81, .(T79, T80), X88) → DELETEC_IN_AGA(T81, T80, X88)

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

DELETEC_IN_AGA(.(T79, T80)) → DELETEC_IN_AGA(T80)

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:

• DELETEC_IN_AGA(.(T79, T80)) → DELETEC_IN_AGA(T80)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

PF_IN_AGGAA(T99, T95, T96, T105, T106) → U11_AGGAA(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
U11_AGGAA(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → PERME_IN_GA(T105, T106)
PERME_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → PF_IN_AGGAA(T99, T95, T96, X106, T100)

The TRS R consists of the following rules:

slowsortA_in_ga([], []) → slowsortA_out_ga([], [])
slowsortA_in_ga(.(T16, .(T17, [])), .(T20, .(T21, []))) → U1_ga(T16, T17, T20, T21, pB_in_aggaa(T20, T16, T17, X22, T21))
pB_in_aggaa(T28, T16, T17, T26, T27) → U7_aggaa(T28, T16, T17, T26, T27, deleteD_in_agga(T28, T16, T17, T26))
deleteD_in_agga(T41, T41, T42, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T57, T55, T56, X59) → U3_agga(T57, T55, T56, X59, deleteC_in_aga(T57, T56, X59))
deleteC_in_aga(T70, .(T70, T71), T71) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(T81, .(T79, T80), X88) → U2_aga(T81, T79, T80, X88, deleteC_in_aga(T81, T80, X88))
U2_aga(T81, T79, T80, X88, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)
U3_agga(T57, T55, T56, X59, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
U7_aggaa(T28, T16, T17, T26, T27, deleteD_out_agga(T28, T16, T17, T26)) → U8_aggaa(T28, T16, T17, T26, T27, pK_in_gag(T26, T27, T28))
pK_in_gag(T26, T86, T28) → U9_gag(T26, T86, T28, permE_in_ga(T26, T86))
permE_in_ga([], []) → permE_out_ga([], [])
permE_in_ga(.(T95, .(T96, [])), .(T99, .(T100, []))) → U4_ga(T95, T96, T99, T100, pF_in_aggaa(T99, T95, T96, X106, T100))
pF_in_aggaa(T99, T95, T96, T105, T106) → U11_aggaa(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
U11_aggaa(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → U12_aggaa(T99, T95, T96, T105, T106, permE_in_ga(T105, T106))
U12_aggaa(T99, T95, T96, T105, T106, permE_out_ga(T105, T106)) → pF_out_aggaa(T99, T95, T96, T105, T106)
U4_ga(T95, T96, T99, T100, pF_out_aggaa(T99, T95, T96, X106, T100)) → permE_out_ga(.(T95, .(T96, [])), .(T99, .(T100, [])))
U9_gag(T26, T86, T28, permE_out_ga(T26, T86)) → U10_gag(T26, T86, T28, sortedH_in_gg(T28, T86))
sortedH_in_gg(T118, T119) → U6_gg(T118, T119, pI_in_gg(T118, T119))
pI_in_gg(T118, T119) → U13_gg(T118, T119, leG_in_gg(T118, T119))
leG_in_gg(s(T132), s(T133)) → U5_gg(T132, T133, leG_in_gg(T132, T133))
leG_in_gg(0, s(T140)) → leG_out_gg(0, s(T140))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T132, T133, leG_out_gg(T132, T133)) → leG_out_gg(s(T132), s(T133))
U13_gg(T118, T119, leG_out_gg(T118, T119)) → U14_gg(T118, T119, sortedJ_in_g(T119))
sortedJ_in_g(T145) → sortedJ_out_g(T145)
U14_gg(T118, T119, sortedJ_out_g(T119)) → pI_out_gg(T118, T119)
U6_gg(T118, T119, pI_out_gg(T118, T119)) → sortedH_out_gg(T118, T119)
U10_gag(T26, T86, T28, sortedH_out_gg(T28, T86)) → pK_out_gag(T26, T86, T28)
U8_aggaa(T28, T16, T17, T26, T27, pK_out_gag(T26, T27, T28)) → pB_out_aggaa(T28, T16, T17, T26, T27)
U1_ga(T16, T17, T20, T21, pB_out_aggaa(T20, T16, T17, X22, T21)) → slowsortA_out_ga(.(T16, .(T17, [])), .(T20, .(T21, [])))

The argument filtering Pi contains the following mapping:
slowsortA_in_ga(x1, x2)  =  slowsortA_in_ga(x1)
[]  =  []
slowsortA_out_ga(x1, x2)  =  slowsortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_aggaa(x1, x2, x3, x4, x5)  =  pB_in_aggaa(x2, x3)
U7_aggaa(x1, x2, x3, x4, x5, x6)  =  U7_aggaa(x2, x3, x6)
deleteD_in_agga(x1, x2, x3, x4)  =  deleteD_in_agga(x2, x3)
deleteD_out_agga(x1, x2, x3, x4)  =  deleteD_out_agga(x1, x2, x3, x4)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x2, x3, x5)
U8_aggaa(x1, x2, x3, x4, x5, x6)  =  U8_aggaa(x1, x2, x3, x4, x6)
pK_in_gag(x1, x2, x3)  =  pK_in_gag(x1, x3)
U9_gag(x1, x2, x3, x4)  =  U9_gag(x1, x3, x4)
permE_in_ga(x1, x2)  =  permE_in_ga(x1)
permE_out_ga(x1, x2)  =  permE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
pF_in_aggaa(x1, x2, x3, x4, x5)  =  pF_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6)  =  U11_aggaa(x2, x3, x6)
U12_aggaa(x1, x2, x3, x4, x5, x6)  =  U12_aggaa(x1, x2, x3, x4, x6)
pF_out_aggaa(x1, x2, x3, x4, x5)  =  pF_out_aggaa(x1, x2, x3, x4, x5)
U10_gag(x1, x2, x3, x4)  =  U10_gag(x1, x2, x3, x4)
sortedH_in_gg(x1, x2)  =  sortedH_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
pI_in_gg(x1, x2)  =  pI_in_gg(x1, x2)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
leG_in_gg(x1, x2)  =  leG_in_gg(x1, x2)
s(x1)  =  s(x1)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
0  =  0
leG_out_gg(x1, x2)  =  leG_out_gg(x1, x2)
U14_gg(x1, x2, x3)  =  U14_gg(x1, x2, x3)
sortedJ_in_g(x1)  =  sortedJ_in_g(x1)
sortedJ_out_g(x1)  =  sortedJ_out_g(x1)
pI_out_gg(x1, x2)  =  pI_out_gg(x1, x2)
sortedH_out_gg(x1, x2)  =  sortedH_out_gg(x1, x2)
pK_out_gag(x1, x2, x3)  =  pK_out_gag(x1, x2, x3)
pB_out_aggaa(x1, x2, x3, x4, x5)  =  pB_out_aggaa(x1, x2, x3, x4, x5)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
PF_IN_AGGAA(x1, x2, x3, x4, x5)  =  PF_IN_AGGAA(x2, x3)
U11_AGGAA(x1, x2, x3, x4, x5, x6)  =  U11_AGGAA(x2, x3, x6)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

PF_IN_AGGAA(T99, T95, T96, T105, T106) → U11_AGGAA(T99, T95, T96, T105, T106, deleteD_in_agga(T99, T95, T96, T105))
U11_AGGAA(T99, T95, T96, T105, T106, deleteD_out_agga(T99, T95, T96, T105)) → PERME_IN_GA(T105, T106)
PERME_IN_GA(.(T95, .(T96, [])), .(T99, .(T100, []))) → PF_IN_AGGAA(T99, T95, T96, X106, T100)

The TRS R consists of the following rules:

deleteD_in_agga(T41, T41, T42, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T57, T55, T56, X59) → U3_agga(T57, T55, T56, X59, deleteC_in_aga(T57, T56, X59))
U3_agga(T57, T55, T56, X59, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
deleteC_in_aga(T70, .(T70, T71), T71) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(T81, .(T79, T80), X88) → U2_aga(T81, T79, T80, X88, deleteC_in_aga(T81, T80, X88))
U2_aga(T81, T79, T80, X88, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
deleteD_in_agga(x1, x2, x3, x4)  =  deleteD_in_agga(x2, x3)
deleteD_out_agga(x1, x2, x3, x4)  =  deleteD_out_agga(x1, x2, x3, x4)
U3_agga(x1, x2, x3, x4, x5)  =  U3_agga(x2, x3, x5)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x2, x3, x5)
PERME_IN_GA(x1, x2)  =  PERME_IN_GA(x1)
PF_IN_AGGAA(x1, x2, x3, x4, x5)  =  PF_IN_AGGAA(x2, x3)
U11_AGGAA(x1, x2, x3, x4, x5, x6)  =  U11_AGGAA(x2, x3, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

PF_IN_AGGAA(T95, T96) → U11_AGGAA(T95, T96, deleteD_in_agga(T95, T96))
U11_AGGAA(T95, T96, deleteD_out_agga(T99, T95, T96, T105)) → PERME_IN_GA(T105)
PERME_IN_GA(.(T95, .(T96, []))) → PF_IN_AGGAA(T95, T96)

The TRS R consists of the following rules:

deleteD_in_agga(T41, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T55, T56) → U3_agga(T55, T56, deleteC_in_aga(T56))
U3_agga(T55, T56, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
deleteC_in_aga(.(T70, T71)) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(.(T79, T80)) → U2_aga(T79, T80, deleteC_in_aga(T80))
U2_aga(T79, T80, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)

The set Q consists of the following terms:

deleteD_in_agga(x0, x1)
U3_agga(x0, x1, x2)
deleteC_in_aga(x0)
U2_aga(x0, x1, x2)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

PERME_IN_GA(.(T95, .(T96, []))) → PF_IN_AGGAA(T95, T96)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂   
POL(PERME_IN_GA(x₁)) = x₁   
POL(PF_IN_AGGAA(x₁, x₂)) = x₂   
POL(U11_AGGAA(x₁, x₂, x₃)) = x₃   
POL(U2_aga(x₁, x₂, x₃)) = x₃   
POL(U3_agga(x₁, x₂, x₃)) = x₃   
POL([]) = 1   
POL(deleteC_in_aga(x₁)) = x₁   
POL(deleteC_out_aga(x₁, x₂, x₃)) = x₃   
POL(deleteD_in_agga(x₁, x₂)) = x₂   
POL(deleteD_out_agga(x₁, x₂, x₃, x₄)) = x₄   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

deleteD_in_agga(T41, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T55, T56) → U3_agga(T55, T56, deleteC_in_aga(T56))
deleteC_in_aga(.(T70, T71)) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(.(T79, T80)) → U2_aga(T79, T80, deleteC_in_aga(T80))
U3_agga(T55, T56, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
U2_aga(T79, T80, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)

(27) Obligation:

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

PF_IN_AGGAA(T95, T96) → U11_AGGAA(T95, T96, deleteD_in_agga(T95, T96))
U11_AGGAA(T95, T96, deleteD_out_agga(T99, T95, T96, T105)) → PERME_IN_GA(T105)

The TRS R consists of the following rules:

deleteD_in_agga(T41, T42) → deleteD_out_agga(T41, T41, T42, T42)
deleteD_in_agga(T55, T56) → U3_agga(T55, T56, deleteC_in_aga(T56))
U3_agga(T55, T56, deleteC_out_aga(T57, T56, X59)) → deleteD_out_agga(T57, T55, T56, X59)
deleteC_in_aga(.(T70, T71)) → deleteC_out_aga(T70, .(T70, T71), T71)
deleteC_in_aga(.(T79, T80)) → U2_aga(T79, T80, deleteC_in_aga(T80))
U2_aga(T79, T80, deleteC_out_aga(T81, T80, X88)) → deleteC_out_aga(T81, .(T79, T80), X88)

The set Q consists of the following terms:

deleteD_in_agga(x0, x1)
U3_agga(x0, x1, x2)
deleteC_in_aga(x0)
U2_aga(x0, x1, x2)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(29) TRUE