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

(0) Obligation:

Clauses:

mergesort([], []).
mergesort(.(E, []), .(E, [])).
mergesort(.(E, .(F, U)), V) :- ','(split(.(E, .(F, U)), W, Y), ','(mergesort(W, X), ','(mergesort(Y, Z), merge(X, Z, V)))).
merge(X, [], X).
merge([], X, X).
merge(.(A, X), .(B, Y), .(A, Z)) :- ','(le(A, B), merge(X, .(B, Y), Z)).
merge(.(A, X), .(B, Y), .(B, Z)) :- ','(gt(A, B), merge(.(A, X), Y, Z)).
split([], [], []).
split(.(E, U), .(E, V), W) :- split(U, W, V).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Query: mergesort(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: mergesort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: mergesort(T1, T2), SCOPE: 1, CLAUSE: 0 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true | mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2) !=? mergesort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true | mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX7 is free\nmergesort(T1, T2) !=? mergesort([], [])\nmergesort(T1, T2) !=? mergesort(.(X7, []), .(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(split(.(T10, .(T11, T12)), X20, X21), ','(mergesort(X20, X22), ','(mergesort(X21, X23), merge(X22, X23, T14))))\n\nT10 is ground\nT11 is ground\nT12 is ground\nX20 is free\nX21 is free\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(split(.(T10, .(T11, T12)), X20, X21), ','(mergesort(X20, X22), ','(mergesort(X21, X23), merge(X22, X23, T14)))), SCOPE: 2, CLAUSE: 7 | ','(split(.(T10, .(T11, T12)), X20, X21), ','(mergesort(X20, X22), ','(mergesort(X21, X23), merge(X22, X23, T14)))), SCOPE: 2, CLAUSE: 8\n\nT10 is ground\nT11 is ground\nT12 is ground\nX20 is free\nX21 is free\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(split(.(T10, .(T11, T12)), X20, X21), ','(mergesort(X20, X22), ','(mergesort(X21, X23), merge(X22, X23, T14)))), SCOPE: 2, CLAUSE: 8\n\nT10 is ground\nT11 is ground\nT12 is ground\nX20 is free\nX21 is free\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="B: ','(split(.(T22, T23), X41, X40), ','(mergesort(.(T21, X40), X22), ','(mergesort(X41, X23), merge(X22, X23, T14))))\n\nT21 is ground\nT22 is ground\nT23 is ground\nX22 is free\nX23 is free\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="D: split(.(T22, T23), X41, X40)\n\nT22 is ground\nT23 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="L: ','(mergesort(.(T21, T25), X22), ','(mergesort(T24, X23), merge(X22, X23, T14)))\n\nT21 is ground\nT24 is ground\nT25 is ground\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: split(.(T22, T23), X41, X40), SCOPE: 3, CLAUSE: 7 | split(.(T22, T23), X41, X40), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: split(.(T22, T23), X41, X40), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground\nX40 is free\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="C: split(T31, X59, X58)\n\nT31 is ground\nX58 is free\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: split(T31, X59, X58), SCOPE: 4, CLAUSE: 7 | split(T31, X59, X58), SCOPE: 4, CLAUSE: 8\n\nT31 is ground\nX58 is free\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: split(T31, X59, X58), SCOPE: 4, CLAUSE: 7\n\nT31 is ground\nX58 is free\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: split(T31, X59, X58), SCOPE: 4, CLAUSE: 8\n\nT31 is ground\nX58 is free\nX59 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: split(T37, X77, X76)\n\nT37 is ground\nX76 is free\nX77 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: mergesort(.(T21, T25), X22)\n\nT21 is ground\nT25 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="M: ','(mergesort(T24, X23), merge(T38, X23, T14))\n\nT24 is ground\nT38 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="E: mergesort(T24, X23)\n\nT24 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="G: merge(T38, T39, T14)\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: mergesort(T24, X23), SCOPE: 5, CLAUSE: 0 | mergesort(T24, X23), SCOPE: 5, CLAUSE: 1 | mergesort(T24, X23), SCOPE: 5, CLAUSE: 2\n\nT24 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: mergesort(T24, X23), SCOPE: 5, CLAUSE: 0\n\nT24 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: mergesort(T24, X23), SCOPE: 5, CLAUSE: 1 | mergesort(T24, X23), SCOPE: 5, CLAUSE: 2\n\nT24 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: mergesort(T24, X23), SCOPE: 5, CLAUSE: 1\n\nT24 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: mergesort(T24, X23), SCOPE: 5, CLAUSE: 2\n\nT24 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
151 [label="151: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="F: ','(split(.(T51, .(T52, T53)), X101, X102), ','(mergesort(X101, X103), ','(mergesort(X102, X104), merge(X103, X104, X105))))\n\nT51 is ground\nT52 is ground\nT53 is ground\nX105 is free\nX101 is free\nX102 is free\nX103 is free\nX104 is free", fontsize=16, style=filled, fillcolor=lightyellow];
156 [label="156: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: split(.(T51, .(T52, T53)), X101, X102)\n\nT51 is ground\nT52 is ground\nT53 is ground\nX101 is free\nX102 is free", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="N: ','(mergesort(T54, X103), ','(mergesort(T55, X104), merge(X103, X104, X105)))\n\nT54 is ground\nT55 is ground\nX105 is free\nX103 is free\nX104 is free", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: mergesort(T54, X103)\n\nT54 is ground\nX103 is free", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="O: ','(mergesort(T55, X104), merge(T56, X104, X105))\n\nT55 is ground\nT56 is ground\nX105 is free\nX104 is free", fontsize=16, style=filled, fillcolor=lightyellow];
184 [label="184: mergesort(T55, X104)\n\nT55 is ground\nX104 is free", fontsize=16, style=filled, fillcolor=lightyellow];
185 [label="185: merge(T56, T57, X105)\n\nT56 is ground\nT57 is ground\nX105 is free", fontsize=16, style=filled, fillcolor=lightyellow];
210 [label="210: merge(T38, T39, T14), SCOPE: 6, CLAUSE: 3 | merge(T38, T39, T14), SCOPE: 6, CLAUSE: 4 | merge(T38, T39, T14), SCOPE: 6, CLAUSE: 5 | merge(T38, T39, T14), SCOPE: 6, CLAUSE: 6\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
211 [label="211: merge(T38, T39, T14), SCOPE: 6, CLAUSE: 3\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="212: merge(T38, T39, T14), SCOPE: 6, CLAUSE: 4 | merge(T38, T39, T14), SCOPE: 6, CLAUSE: 5 | merge(T38, T39, T14), SCOPE: 6, CLAUSE: 6\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="213: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
217 [label="217: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: merge(T38, T39, T14), SCOPE: 6, CLAUSE: 4\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: merge(T38, T39, T14), SCOPE: 6, CLAUSE: 5 | merge(T38, T39, T14), SCOPE: 6, CLAUSE: 6\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
224 [label="224: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
229 [label="229: merge(T38, T39, T14), SCOPE: 6, CLAUSE: 5\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
230 [label="230: merge(T38, T39, T14), SCOPE: 6, CLAUSE: 6\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="H: ','(le(T92, T94), merge(T93, .(T94, T95), T97))\n\nT92 is ground\nT93 is ground\nT94 is ground\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
233 [label="J: le(T92, T94)\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
234 [label="234: merge(T93, .(T94, T95), T97)\n\nT93 is ground\nT94 is ground\nT95 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
237 [label="237: le(T92, T94), SCOPE: 7, CLAUSE: 11 | le(T92, T94), SCOPE: 7, CLAUSE: 12 | le(T92, T94), SCOPE: 7, CLAUSE: 13\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="238: le(T92, T94), SCOPE: 7, CLAUSE: 11\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
239 [label="239: le(T92, T94), SCOPE: 7, CLAUSE: 12 | le(T92, T94), SCOPE: 7, CLAUSE: 13\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
242 [label="242: le(T110, T111)\n\nT110 is ground\nT111 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
243 [label="243: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
250 [label="250: le(T92, T94), SCOPE: 7, CLAUSE: 12\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
251 [label="251: le(T92, T94), SCOPE: 7, CLAUSE: 13\n\nT92 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
252 [label="252: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
253 [label="253: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
254 [label="254: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
255 [label="255: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
256 [label="256: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
257 [label="257: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
258 [label="I: ','(gt(T131, T133), merge(.(T131, T132), T134, T136))\n\nT131 is ground\nT132 is ground\nT133 is ground\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
259 [label="259: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
260 [label="K: gt(T131, T133)\n\nT131 is ground\nT133 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
261 [label="261: merge(.(T131, T132), T134, T136)\n\nT131 is ground\nT132 is ground\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
264 [label="264: gt(T131, T133), SCOPE: 8, CLAUSE: 9 | gt(T131, T133), SCOPE: 8, CLAUSE: 10\n\nT131 is ground\nT133 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
265 [label="265: gt(T131, T133), SCOPE: 8, CLAUSE: 9\n\nT131 is ground\nT133 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
266 [label="266: gt(T131, T133), SCOPE: 8, CLAUSE: 10\n\nT131 is ground\nT133 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
269 [label="269: gt(T149, T150)\n\nT149 is ground\nT150 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
270 [label="270: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
271 [label="271: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
272 [label="272: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
273 [label="273: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
274 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 274 [arrowhead = none ];
274 -> 4;

275 [label="EVAL with clause\nmergesort([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 275 [arrowhead = none ];
275 -> 13;

276 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 276 [arrowhead = none ];
276 -> 14;

277 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 277 [arrowhead = none ];
277 -> 15;

278 [label="EVAL with clause\nmergesort(.(X7, []), .(X7, [])).\nand substitutionX7 -> T4,\nT1 -> .(T4, []),\nT2 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 278 [arrowhead = none ];
278 -> 23;

279 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 279 [arrowhead = none ];
279 -> 27;

280 [label="BACKTRACK\nfor clause: mergesort(.(E, []), .(E, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 280 [arrowhead = none ];
280 -> 18;

281 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(.(E, .(F, U)), W, Y), ','(mergesort(W, X), ','(mergesort(Y, Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 281 [arrowhead = none ];
281 -> 20;

282 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 282 [arrowhead = none ];
282 -> 29;

283 [label="EVAL with clause\nmergesort(.(X16, .(X17, X18)), X19) :- ','(split(.(X16, .(X17, X18)), X20, X21), ','(mergesort(X20, X22), ','(mergesort(X21, X23), merge(X22, X23, X19)))).\nand substitutionX16 -> T10,\nX17 -> T11,\nX18 -> T12,\nT1 -> .(T10, .(T11, T12)),\nT2 -> T14,\nX19 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 283 [arrowhead = none ];
283 -> 35;

284 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 284 [arrowhead = none ];
284 -> 37;

285 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(.(E, .(F, U)), W, Y), ','(mergesort(W, X), ','(mergesort(Y, Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 285 [arrowhead = none ];
285 -> 33;

286 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 286 [arrowhead = none ];
286 -> 41;

287 [label="BACKTRACK\nfor clause: split([], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 287 [arrowhead = none ];
287 -> 42;

288 [label="ONLY EVAL with clause\nsplit(.(X36, X37), .(X36, X38), X39) :- split(X37, X39, X38).\nand substitutionT10 -> T21,\nX36 -> T21,\nT11 -> T22,\nT12 -> T23,\nX37 -> .(T22, T23),\nX38 -> X40,\nX20 -> .(T21, X40),\nX21 -> X41,\nX39 -> X41", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 288 [arrowhead = none ];
288 -> 50;

289 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 289 [arrowhead = none ];
289 -> 51;

290 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nT23 is ground\nT24 is ground\nT25 is ground\nreplacements:X41 -> T24,\nX40 -> T25", fontsize=14, style = filled, fillcolor = violet];
50 -> 290 [arrowhead = none ];
290 -> 52;

291 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 291 [arrowhead = none ];
291 -> 57;

292 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 292 [arrowhead = none ];
292 -> 114;

293 [label="SPLIT 2\nnew knowledge:\nT21 is ground\nT25 is ground\nT38 is ground\nreplacements:X22 -> T38", fontsize=14, style = filled, fillcolor = violet];
52 -> 293 [arrowhead = none ];
293 -> 115;

294 [label="BACKTRACK\nfor clause: split([], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 294 [arrowhead = none ];
294 -> 58;

295 [label="ONLY EVAL with clause\nsplit(.(X54, X55), .(X54, X56), X57) :- split(X55, X57, X56).\nand substitutionT22 -> T30,\nX54 -> T30,\nT23 -> T31,\nX55 -> T31,\nX56 -> X58,\nX41 -> .(T30, X58),\nX40 -> X59,\nX57 -> X59", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 295 [arrowhead = none ];
295 -> 69;

296 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 296 [arrowhead = none ];
296 -> 70;

297 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
70 -> 297 [arrowhead = none ];
297 -> 71;

298 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
70 -> 298 [arrowhead = none ];
298 -> 72;

299 [label="EVAL with clause\nsplit([], [], []).\nand substitutionT31 -> [],\nX59 -> [],\nX58 -> []", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 299 [arrowhead = none ];
299 -> 73;

300 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 300 [arrowhead = none ];
300 -> 74;

301 [label="EVAL with clause\nsplit(.(X72, X73), .(X72, X74), X75) :- split(X73, X75, X74).\nand substitutionX72 -> T36,\nX73 -> T37,\nT31 -> .(T36, T37),\nX74 -> X76,\nX59 -> .(T36, X76),\nX58 -> X77,\nX75 -> X77", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 301 [arrowhead = none ];
301 -> 90;

302 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 302 [arrowhead = none ];
302 -> 92;

303 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 303 [arrowhead = none ];
303 -> 75;

304 [label="INSTANCE with matching:\nT31 -> T37\nX59 -> X77\nX58 -> X76", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 304 [arrowhead = none , style=dashed];
304 -> 69[style=dashed];

305 [label="INSTANCE with matching:\nT1 -> .(T21, T25)\nT2 -> X22", fontsize=14, style = filled, fillcolor = lightgrey];
114 -> 305 [arrowhead = none , style=dashed];
305 -> 1[style=dashed];

306 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
115 -> 306 [arrowhead = none ];
306 -> 118;

307 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT39 is ground\nreplacements:X23 -> T39", fontsize=14, style = filled, fillcolor = violet];
115 -> 307 [arrowhead = none ];
307 -> 119;

308 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
118 -> 308 [arrowhead = none ];
308 -> 120;

309 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
119 -> 309 [arrowhead = none ];
309 -> 210;

310 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
120 -> 310 [arrowhead = none ];
310 -> 121;

311 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
120 -> 311 [arrowhead = none ];
311 -> 122;

312 [label="EVAL with clause\nmergesort([], []).\nand substitutionT24 -> [],\nX23 -> []", fontsize=14, style = filled, fillcolor = lightblue];
121 -> 312 [arrowhead = none ];
312 -> 123;

313 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
121 -> 313 [arrowhead = none ];
313 -> 126;

314 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 314 [arrowhead = none ];
314 -> 128;

315 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
122 -> 315 [arrowhead = none ];
315 -> 129;

316 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
123 -> 316 [arrowhead = none ];
316 -> 127;

317 [label="EVAL with clause\nmergesort(.(X82, []), .(X82, [])).\nand substitutionX82 -> T44,\nT24 -> .(T44, []),\nX23 -> .(T44, [])", fontsize=14, style = filled, fillcolor = lightblue];
128 -> 317 [arrowhead = none ];
317 -> 130;

318 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 318 [arrowhead = none ];
318 -> 131;

319 [label="EVAL with clause\nmergesort(.(X97, .(X98, X99)), X100) :- ','(split(.(X97, .(X98, X99)), X101, X102), ','(mergesort(X101, X103), ','(mergesort(X102, X104), merge(X103, X104, X100)))).\nand substitutionX97 -> T51,\nX98 -> T52,\nX99 -> T53,\nT24 -> .(T51, .(T52, T53)),\nX23 -> X105,\nX100 -> X105", fontsize=14, style = filled, fillcolor = lightblue];
129 -> 319 [arrowhead = none ];
319 -> 155;

320 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
129 -> 320 [arrowhead = none ];
320 -> 156;

321 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
130 -> 321 [arrowhead = none ];
321 -> 151;

322 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
155 -> 322 [arrowhead = none ];
322 -> 159;

323 [label="SPLIT 2\nnew knowledge:\nT51 is ground\nT52 is ground\nT53 is ground\nT54 is ground\nT55 is ground\nreplacements:X101 -> T54,\nX102 -> T55", fontsize=14, style = filled, fillcolor = violet];
155 -> 323 [arrowhead = none ];
323 -> 160;

324 [label="INSTANCE with matching:\nT22 -> T51\nT23 -> .(T52, T53)\nX41 -> X101\nX40 -> X102", fontsize=14, style = filled, fillcolor = lightgrey];
159 -> 324 [arrowhead = none , style=dashed];
324 -> 51[style=dashed];

325 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
160 -> 325 [arrowhead = none ];
325 -> 174;

326 [label="SPLIT 2\nnew knowledge:\nT54 is ground\nT56 is ground\nreplacements:X103 -> T56", fontsize=14, style = filled, fillcolor = violet];
160 -> 326 [arrowhead = none ];
326 -> 175;

327 [label="INSTANCE with matching:\nT24 -> T54\nX23 -> X103", fontsize=14, style = filled, fillcolor = lightgrey];
174 -> 327 [arrowhead = none , style=dashed];
327 -> 118[style=dashed];

328 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
175 -> 328 [arrowhead = none ];
328 -> 184;

329 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nT57 is ground\nreplacements:X104 -> T57", fontsize=14, style = filled, fillcolor = violet];
175 -> 329 [arrowhead = none ];
329 -> 185;

330 [label="INSTANCE with matching:\nT24 -> T55\nX23 -> X104", fontsize=14, style = filled, fillcolor = lightgrey];
184 -> 330 [arrowhead = none , style=dashed];
330 -> 118[style=dashed];

331 [label="INSTANCE with matching:\nT38 -> T56\nT39 -> T57\nT14 -> X105", fontsize=14, style = filled, fillcolor = lightgrey];
185 -> 331 [arrowhead = none , style=dashed];
331 -> 119[style=dashed];

332 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
210 -> 332 [arrowhead = none ];
332 -> 211;

333 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
210 -> 333 [arrowhead = none ];
333 -> 212;

334 [label="EVAL with clause\nmerge(X114, [], X114).\nand substitutionT38 -> T66,\nX114 -> T66,\nT39 -> [],\nT14 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
211 -> 334 [arrowhead = none ];
334 -> 213;

335 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
211 -> 335 [arrowhead = none ];
335 -> 215;

336 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
212 -> 336 [arrowhead = none ];
336 -> 221;

337 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
212 -> 337 [arrowhead = none ];
337 -> 222;

338 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
213 -> 338 [arrowhead = none ];
338 -> 217;

339 [label="EVAL with clause\nmerge([], X119, X119).\nand substitutionT38 -> [],\nT39 -> T71,\nX119 -> T71,\nT14 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
221 -> 339 [arrowhead = none ];
339 -> 223;

340 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
221 -> 340 [arrowhead = none ];
340 -> 224;

341 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
222 -> 341 [arrowhead = none ];
341 -> 229;

342 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
222 -> 342 [arrowhead = none ];
342 -> 230;

343 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
223 -> 343 [arrowhead = none ];
343 -> 226;

344 [label="EVAL with clause\nmerge(.(X140, X141), .(X142, X143), .(X140, X144)) :- ','(le(X140, X142), merge(X141, .(X142, X143), X144)).\nand substitutionX140 -> T92,\nX141 -> T93,\nT38 -> .(T92, T93),\nX142 -> T94,\nX143 -> T95,\nT39 -> .(T94, T95),\nX144 -> T97,\nT14 -> .(T92, T97),\nT96 -> T97", fontsize=14, style = filled, fillcolor = lightblue];
229 -> 344 [arrowhead = none ];
344 -> 231;

345 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
229 -> 345 [arrowhead = none ];
345 -> 232;

346 [label="EVAL with clause\nmerge(.(X178, X179), .(X180, X181), .(X180, X182)) :- ','(gt(X178, X180), merge(.(X178, X179), X181, X182)).\nand substitutionX178 -> T131,\nX179 -> T132,\nT38 -> .(T131, T132),\nX180 -> T133,\nX181 -> T134,\nT39 -> .(T133, T134),\nX182 -> T136,\nT14 -> .(T133, T136),\nT135 -> T136", fontsize=14, style = filled, fillcolor = lightblue];
230 -> 346 [arrowhead = none ];
346 -> 258;

347 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
230 -> 347 [arrowhead = none ];
347 -> 259;

348 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
231 -> 348 [arrowhead = none ];
348 -> 233;

349 [label="SPLIT 2\nnew knowledge:\nT92 is ground\nT94 is ground", fontsize=14, style = filled, fillcolor = violet];
231 -> 349 [arrowhead = none ];
349 -> 234;

350 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
233 -> 350 [arrowhead = none ];
350 -> 237;

351 [label="INSTANCE with matching:\nT38 -> T93\nT39 -> .(T94, T95)\nT14 -> T97", fontsize=14, style = filled, fillcolor = lightgrey];
234 -> 351 [arrowhead = none , style=dashed];
351 -> 119[style=dashed];

352 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
237 -> 352 [arrowhead = none ];
352 -> 238;

353 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
237 -> 353 [arrowhead = none ];
353 -> 239;

354 [label="EVAL with clause\nle(s(X157), s(X158)) :- le(X157, X158).\nand substitutionX157 -> T110,\nT92 -> s(T110),\nX158 -> T111,\nT94 -> s(T111)", fontsize=14, style = filled, fillcolor = lightblue];
238 -> 354 [arrowhead = none ];
354 -> 242;

355 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
238 -> 355 [arrowhead = none ];
355 -> 243;

356 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
239 -> 356 [arrowhead = none ];
356 -> 250;

357 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
239 -> 357 [arrowhead = none ];
357 -> 251;

358 [label="INSTANCE with matching:\nT92 -> T110\nT94 -> T111", fontsize=14, style = filled, fillcolor = lightgrey];
242 -> 358 [arrowhead = none , style=dashed];
358 -> 233[style=dashed];

359 [label="EVAL with clause\nle(0, s(X165)).\nand substitutionT92 -> 0,\nX165 -> T118,\nT94 -> s(T118)", fontsize=14, style = filled, fillcolor = lightblue];
250 -> 359 [arrowhead = none ];
359 -> 252;

360 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
250 -> 360 [arrowhead = none ];
360 -> 253;

361 [label="EVAL with clause\nle(0, 0).\nand substitutionT92 -> 0,\nT94 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
251 -> 361 [arrowhead = none ];
361 -> 255;

362 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
251 -> 362 [arrowhead = none ];
362 -> 256;

363 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
252 -> 363 [arrowhead = none ];
363 -> 254;

364 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
255 -> 364 [arrowhead = none ];
364 -> 257;

365 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
258 -> 365 [arrowhead = none ];
365 -> 260;

366 [label="SPLIT 2\nnew knowledge:\nT131 is ground\nT133 is ground", fontsize=14, style = filled, fillcolor = violet];
258 -> 366 [arrowhead = none ];
366 -> 261;

367 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
260 -> 367 [arrowhead = none ];
367 -> 264;

368 [label="INSTANCE with matching:\nT38 -> .(T131, T132)\nT39 -> T134\nT14 -> T136", fontsize=14, style = filled, fillcolor = lightgrey];
261 -> 368 [arrowhead = none , style=dashed];
368 -> 119[style=dashed];

369 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
264 -> 369 [arrowhead = none ];
369 -> 265;

370 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
264 -> 370 [arrowhead = none ];
370 -> 266;

371 [label="EVAL with clause\ngt(s(X195), s(X196)) :- gt(X195, X196).\nand substitutionX195 -> T149,\nT131 -> s(T149),\nX196 -> T150,\nT133 -> s(T150)", fontsize=14, style = filled, fillcolor = lightblue];
265 -> 371 [arrowhead = none ];
371 -> 269;

372 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
265 -> 372 [arrowhead = none ];
372 -> 270;

373 [label="EVAL with clause\ngt(s(X201), 0).\nand substitutionX201 -> T155,\nT131 -> s(T155),\nT133 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
266 -> 373 [arrowhead = none ];
373 -> 271;

374 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
266 -> 374 [arrowhead = none ];
374 -> 272;

375 [label="INSTANCE with matching:\nT131 -> T149\nT133 -> T150", fontsize=14, style = filled, fillcolor = lightgrey];
269 -> 375 [arrowhead = none , style=dashed];
375 -> 260[style=dashed];

376 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
271 -> 376 [arrowhead = none ];
376 -> 273;

}

(2) Obligation:

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

mergesortA_in_ga([], []) → mergesortA_out_ga([], [])
mergesortA_in_ga(.(T4, []), .(T4, [])) → mergesortA_out_ga(.(T4, []), .(T4, []))
mergesortA_in_ga(.(T21, .(T22, T23)), T14) → U1_ga(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
pB_in_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14) → U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
pL_in_ggagaa(T21, T25, T38, T24, X23, T14) → U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
pM_in_gaga(T24, T39, T38, T14) → U13_gaga(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
U13_gaga(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_gaga(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U14_gaga(T24, T39, T38, T14, mergeG_out_gga(T38, T39, T14)) → pM_out_gaga(T24, T39, T38, T14)
U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_out_gaga(T24, X23, T38, T14)) → pL_out_ggagaa(T21, T25, T38, T24, X23, T14)
U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_out_ggagaa(T21, T25, X22, T24, X23, T14)) → pB_out_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14)
U1_ga(T21, T22, T23, T14, pB_out_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14)) → mergesortA_out_ga(.(T21, .(T22, T23)), T14)

The argument filtering Pi contains the following mapping:
mergesortA_in_ga(x1, x2)  =  mergesortA_in_ga(x1)
[]  =  []
mergesortA_out_ga(x1, x2)  =  mergesortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaagaaa(x1, x2, x5)
U9_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_ggaagaaa(x1, x2, x5, x9)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
U10_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_ggaagaaa(x1, x2, x3, x4, x5, x9)
pL_in_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_in_ggagaa(x1, x2, x4)
U11_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_ggagaa(x1, x2, x4, x7)
U12_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggagaa(x1, x2, x3, x4, x7)
pM_in_gaga(x1, x2, x3, x4)  =  pM_in_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5)  =  U13_gaga(x1, x3, x5)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U14_gaga(x1, x2, x3, x4, x5)  =  U14_gaga(x1, x2, x3, x5)
pM_out_gaga(x1, x2, x3, x4)  =  pM_out_gaga(x1, x2, x3, x4)
pL_out_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_out_ggagaa(x1, x2, x3, x4, x5, x6)
pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)

(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:

MERGESORTA_IN_GA(.(T21, .(T22, T23)), T14) → U1_GA(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
MERGESORTA_IN_GA(.(T21, .(T22, T23)), T14) → PB_IN_GGAAGAAA(T22, T23, X41, X40, T21, X22, X23, T14)
PB_IN_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14) → U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
PB_IN_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14) → SPLITD_IN_GGAA(T22, T23, T24, T25)
SPLITD_IN_GGAA(T30, T31, .(T30, X58), X59) → U3_GGAA(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
SPLITD_IN_GGAA(T30, T31, .(T30, X58), X59) → SPLITC_IN_GAA(T31, X59, X58)
SPLITC_IN_GAA(.(T36, T37), .(T36, X76), X77) → U2_GAA(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
SPLITC_IN_GAA(.(T36, T37), .(T36, X76), X77) → SPLITC_IN_GAA(T37, X77, X76)
U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → PL_IN_GGAGAA(T21, T25, X22, T24, X23, T14)
PL_IN_GGAGAA(T21, T25, T38, T24, X23, T14) → U11_GGAGAA(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
PL_IN_GGAGAA(T21, T25, T38, T24, X23, T14) → MERGESORTA_IN_GA(.(T21, T25), T38)
U11_GGAGAA(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_GGAGAA(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
U11_GGAGAA(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → PM_IN_GAGA(T24, X23, T38, T14)
PM_IN_GAGA(T24, T39, T38, T14) → U13_GAGA(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
PM_IN_GAGA(T24, T39, T38, T14) → MERGESORTE_IN_GA(T24, T39)
MERGESORTE_IN_GA(.(T51, .(T52, T53)), X105) → U4_GA(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
MERGESORTE_IN_GA(.(T51, .(T52, T53)), X105) → PF_IN_GGGAAAAA(T51, T52, T53, X101, X102, X103, X104, X105)
PF_IN_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105) → U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
PF_IN_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105) → SPLITD_IN_GGAA(T51, .(T52, T53), T54, T55)
U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → PN_IN_GAGAA(T54, X103, T55, X104, X105)
PN_IN_GAGAA(T54, T56, T55, X104, X105) → U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
PN_IN_GAGAA(T54, T56, T55, X104, X105) → MERGESORTE_IN_GA(T54, T56)
U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_GAGAA(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → PO_IN_GAGA(T55, X104, T56, X105)
PO_IN_GAGA(T55, T57, T56, X105) → U19_GAGA(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
PO_IN_GAGA(T55, T57, T56, X105) → MERGESORTE_IN_GA(T55, T57)
U19_GAGA(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_GAGA(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
U19_GAGA(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → MERGEG_IN_GGA(T56, T57, X105)
MERGEG_IN_GGA(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_GGA(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
MERGEG_IN_GGA(.(T92, T93), .(T94, T95), .(T92, T97)) → PH_IN_GGGGA(T92, T94, T93, T95, T97)
PH_IN_GGGGA(T92, T94, T93, T95, T97) → U21_GGGGA(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
PH_IN_GGGGA(T92, T94, T93, T95, T97) → LEJ_IN_GG(T92, T94)
LEJ_IN_GG(s(T110), s(T111)) → U7_GG(T110, T111, leJ_in_gg(T110, T111))
LEJ_IN_GG(s(T110), s(T111)) → LEJ_IN_GG(T110, T111)
U21_GGGGA(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_GGGGA(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
U21_GGGGA(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → MERGEG_IN_GGA(T93, .(T94, T95), T97)
MERGEG_IN_GGA(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_GGA(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
MERGEG_IN_GGA(.(T131, T132), .(T133, T134), .(T133, T136)) → PI_IN_GGGGA(T131, T133, T132, T134, T136)
PI_IN_GGGGA(T131, T133, T132, T134, T136) → U23_GGGGA(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
PI_IN_GGGGA(T131, T133, T132, T134, T136) → GTK_IN_GG(T131, T133)
GTK_IN_GG(s(T149), s(T150)) → U8_GG(T149, T150, gtK_in_gg(T149, T150))
GTK_IN_GG(s(T149), s(T150)) → GTK_IN_GG(T149, T150)
U23_GGGGA(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_GGGGA(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U23_GGGGA(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → MERGEG_IN_GGA(.(T131, T132), T134, T136)
U13_GAGA(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_GAGA(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U13_GAGA(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → MERGEG_IN_GGA(T38, T39, T14)

The TRS R consists of the following rules:

mergesortA_in_ga([], []) → mergesortA_out_ga([], [])
mergesortA_in_ga(.(T4, []), .(T4, [])) → mergesortA_out_ga(.(T4, []), .(T4, []))
mergesortA_in_ga(.(T21, .(T22, T23)), T14) → U1_ga(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
pB_in_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14) → U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
pL_in_ggagaa(T21, T25, T38, T24, X23, T14) → U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
pM_in_gaga(T24, T39, T38, T14) → U13_gaga(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
U13_gaga(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_gaga(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U14_gaga(T24, T39, T38, T14, mergeG_out_gga(T38, T39, T14)) → pM_out_gaga(T24, T39, T38, T14)
U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_out_gaga(T24, X23, T38, T14)) → pL_out_ggagaa(T21, T25, T38, T24, X23, T14)
U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_out_ggagaa(T21, T25, X22, T24, X23, T14)) → pB_out_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14)
U1_ga(T21, T22, T23, T14, pB_out_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14)) → mergesortA_out_ga(.(T21, .(T22, T23)), T14)

The argument filtering Pi contains the following mapping:
mergesortA_in_ga(x1, x2)  =  mergesortA_in_ga(x1)
[]  =  []
mergesortA_out_ga(x1, x2)  =  mergesortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaagaaa(x1, x2, x5)
U9_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_ggaagaaa(x1, x2, x5, x9)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
U10_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_ggaagaaa(x1, x2, x3, x4, x5, x9)
pL_in_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_in_ggagaa(x1, x2, x4)
U11_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_ggagaa(x1, x2, x4, x7)
U12_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggagaa(x1, x2, x3, x4, x7)
pM_in_gaga(x1, x2, x3, x4)  =  pM_in_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5)  =  U13_gaga(x1, x3, x5)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U14_gaga(x1, x2, x3, x4, x5)  =  U14_gaga(x1, x2, x3, x5)
pM_out_gaga(x1, x2, x3, x4)  =  pM_out_gaga(x1, x2, x3, x4)
pL_out_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_out_ggagaa(x1, x2, x3, x4, x5, x6)
pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
MERGESORTA_IN_GA(x1, x2)  =  MERGESORTA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
PB_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGAAGAAA(x1, x2, x5)
U9_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_GGAAGAAA(x1, x2, x5, x9)
SPLITD_IN_GGAA(x1, x2, x3, x4)  =  SPLITD_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5)  =  U3_GGAA(x1, x2, x5)
SPLITC_IN_GAA(x1, x2, x3)  =  SPLITC_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x2, x5)
U10_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_GGAAGAAA(x1, x2, x3, x4, x5, x9)
PL_IN_GGAGAA(x1, x2, x3, x4, x5, x6)  =  PL_IN_GGAGAA(x1, x2, x4)
U11_GGAGAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_GGAGAA(x1, x2, x4, x7)
U12_GGAGAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GGAGAA(x1, x2, x3, x4, x7)
PM_IN_GAGA(x1, x2, x3, x4)  =  PM_IN_GAGA(x1, x3)
U13_GAGA(x1, x2, x3, x4, x5)  =  U13_GAGA(x1, x3, x5)
MERGESORTE_IN_GA(x1, x2)  =  MERGESORTE_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
PF_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PF_IN_GGGAAAAA(x1, x2, x3)
U15_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_GGGAAAAA(x1, x2, x3, x9)
U16_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGAAAAA(x1, x2, x3, x4, x5, x9)
PN_IN_GAGAA(x1, x2, x3, x4, x5)  =  PN_IN_GAGAA(x1, x3)
U17_GAGAA(x1, x2, x3, x4, x5, x6)  =  U17_GAGAA(x1, x3, x6)
U18_GAGAA(x1, x2, x3, x4, x5, x6)  =  U18_GAGAA(x1, x2, x3, x6)
PO_IN_GAGA(x1, x2, x3, x4)  =  PO_IN_GAGA(x1, x3)
U19_GAGA(x1, x2, x3, x4, x5)  =  U19_GAGA(x1, x3, x5)
U20_GAGA(x1, x2, x3, x4, x5)  =  U20_GAGA(x1, x2, x3, x5)
MERGEG_IN_GGA(x1, x2, x3)  =  MERGEG_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
PH_IN_GGGGA(x1, x2, x3, x4, x5)  =  PH_IN_GGGGA(x1, x2, x3, x4)
U21_GGGGA(x1, x2, x3, x4, x5, x6)  =  U21_GGGGA(x1, x2, x3, x4, x6)
LEJ_IN_GG(x1, x2)  =  LEJ_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U22_GGGGA(x1, x2, x3, x4, x5, x6)  =  U22_GGGGA(x1, x2, x3, x4, x6)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)
PI_IN_GGGGA(x1, x2, x3, x4, x5)  =  PI_IN_GGGGA(x1, x2, x3, x4)
U23_GGGGA(x1, x2, x3, x4, x5, x6)  =  U23_GGGGA(x1, x2, x3, x4, x6)
GTK_IN_GG(x1, x2)  =  GTK_IN_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)
U24_GGGGA(x1, x2, x3, x4, x5, x6)  =  U24_GGGGA(x1, x2, x3, x4, x6)
U14_GAGA(x1, x2, x3, x4, x5)  =  U14_GAGA(x1, x2, x3, x5)

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

(4) Obligation:

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

MERGESORTA_IN_GA(.(T21, .(T22, T23)), T14) → U1_GA(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
MERGESORTA_IN_GA(.(T21, .(T22, T23)), T14) → PB_IN_GGAAGAAA(T22, T23, X41, X40, T21, X22, X23, T14)
PB_IN_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14) → U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
PB_IN_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14) → SPLITD_IN_GGAA(T22, T23, T24, T25)
SPLITD_IN_GGAA(T30, T31, .(T30, X58), X59) → U3_GGAA(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
SPLITD_IN_GGAA(T30, T31, .(T30, X58), X59) → SPLITC_IN_GAA(T31, X59, X58)
SPLITC_IN_GAA(.(T36, T37), .(T36, X76), X77) → U2_GAA(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
SPLITC_IN_GAA(.(T36, T37), .(T36, X76), X77) → SPLITC_IN_GAA(T37, X77, X76)
U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → PL_IN_GGAGAA(T21, T25, X22, T24, X23, T14)
PL_IN_GGAGAA(T21, T25, T38, T24, X23, T14) → U11_GGAGAA(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
PL_IN_GGAGAA(T21, T25, T38, T24, X23, T14) → MERGESORTA_IN_GA(.(T21, T25), T38)
U11_GGAGAA(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_GGAGAA(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
U11_GGAGAA(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → PM_IN_GAGA(T24, X23, T38, T14)
PM_IN_GAGA(T24, T39, T38, T14) → U13_GAGA(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
PM_IN_GAGA(T24, T39, T38, T14) → MERGESORTE_IN_GA(T24, T39)
MERGESORTE_IN_GA(.(T51, .(T52, T53)), X105) → U4_GA(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
MERGESORTE_IN_GA(.(T51, .(T52, T53)), X105) → PF_IN_GGGAAAAA(T51, T52, T53, X101, X102, X103, X104, X105)
PF_IN_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105) → U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
PF_IN_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105) → SPLITD_IN_GGAA(T51, .(T52, T53), T54, T55)
U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → PN_IN_GAGAA(T54, X103, T55, X104, X105)
PN_IN_GAGAA(T54, T56, T55, X104, X105) → U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
PN_IN_GAGAA(T54, T56, T55, X104, X105) → MERGESORTE_IN_GA(T54, T56)
U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_GAGAA(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → PO_IN_GAGA(T55, X104, T56, X105)
PO_IN_GAGA(T55, T57, T56, X105) → U19_GAGA(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
PO_IN_GAGA(T55, T57, T56, X105) → MERGESORTE_IN_GA(T55, T57)
U19_GAGA(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_GAGA(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
U19_GAGA(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → MERGEG_IN_GGA(T56, T57, X105)
MERGEG_IN_GGA(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_GGA(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
MERGEG_IN_GGA(.(T92, T93), .(T94, T95), .(T92, T97)) → PH_IN_GGGGA(T92, T94, T93, T95, T97)
PH_IN_GGGGA(T92, T94, T93, T95, T97) → U21_GGGGA(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
PH_IN_GGGGA(T92, T94, T93, T95, T97) → LEJ_IN_GG(T92, T94)
LEJ_IN_GG(s(T110), s(T111)) → U7_GG(T110, T111, leJ_in_gg(T110, T111))
LEJ_IN_GG(s(T110), s(T111)) → LEJ_IN_GG(T110, T111)
U21_GGGGA(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_GGGGA(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
U21_GGGGA(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → MERGEG_IN_GGA(T93, .(T94, T95), T97)
MERGEG_IN_GGA(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_GGA(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
MERGEG_IN_GGA(.(T131, T132), .(T133, T134), .(T133, T136)) → PI_IN_GGGGA(T131, T133, T132, T134, T136)
PI_IN_GGGGA(T131, T133, T132, T134, T136) → U23_GGGGA(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
PI_IN_GGGGA(T131, T133, T132, T134, T136) → GTK_IN_GG(T131, T133)
GTK_IN_GG(s(T149), s(T150)) → U8_GG(T149, T150, gtK_in_gg(T149, T150))
GTK_IN_GG(s(T149), s(T150)) → GTK_IN_GG(T149, T150)
U23_GGGGA(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_GGGGA(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U23_GGGGA(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → MERGEG_IN_GGA(.(T131, T132), T134, T136)
U13_GAGA(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_GAGA(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U13_GAGA(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → MERGEG_IN_GGA(T38, T39, T14)

The TRS R consists of the following rules:

mergesortA_in_ga([], []) → mergesortA_out_ga([], [])
mergesortA_in_ga(.(T4, []), .(T4, [])) → mergesortA_out_ga(.(T4, []), .(T4, []))
mergesortA_in_ga(.(T21, .(T22, T23)), T14) → U1_ga(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
pB_in_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14) → U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
pL_in_ggagaa(T21, T25, T38, T24, X23, T14) → U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
pM_in_gaga(T24, T39, T38, T14) → U13_gaga(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
U13_gaga(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_gaga(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U14_gaga(T24, T39, T38, T14, mergeG_out_gga(T38, T39, T14)) → pM_out_gaga(T24, T39, T38, T14)
U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_out_gaga(T24, X23, T38, T14)) → pL_out_ggagaa(T21, T25, T38, T24, X23, T14)
U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_out_ggagaa(T21, T25, X22, T24, X23, T14)) → pB_out_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14)
U1_ga(T21, T22, T23, T14, pB_out_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14)) → mergesortA_out_ga(.(T21, .(T22, T23)), T14)

The argument filtering Pi contains the following mapping:
mergesortA_in_ga(x1, x2)  =  mergesortA_in_ga(x1)
[]  =  []
mergesortA_out_ga(x1, x2)  =  mergesortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaagaaa(x1, x2, x5)
U9_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_ggaagaaa(x1, x2, x5, x9)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
U10_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_ggaagaaa(x1, x2, x3, x4, x5, x9)
pL_in_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_in_ggagaa(x1, x2, x4)
U11_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_ggagaa(x1, x2, x4, x7)
U12_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggagaa(x1, x2, x3, x4, x7)
pM_in_gaga(x1, x2, x3, x4)  =  pM_in_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5)  =  U13_gaga(x1, x3, x5)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U14_gaga(x1, x2, x3, x4, x5)  =  U14_gaga(x1, x2, x3, x5)
pM_out_gaga(x1, x2, x3, x4)  =  pM_out_gaga(x1, x2, x3, x4)
pL_out_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_out_ggagaa(x1, x2, x3, x4, x5, x6)
pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
MERGESORTA_IN_GA(x1, x2)  =  MERGESORTA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
PB_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGAAGAAA(x1, x2, x5)
U9_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_GGAAGAAA(x1, x2, x5, x9)
SPLITD_IN_GGAA(x1, x2, x3, x4)  =  SPLITD_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5)  =  U3_GGAA(x1, x2, x5)
SPLITC_IN_GAA(x1, x2, x3)  =  SPLITC_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x2, x5)
U10_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_GGAAGAAA(x1, x2, x3, x4, x5, x9)
PL_IN_GGAGAA(x1, x2, x3, x4, x5, x6)  =  PL_IN_GGAGAA(x1, x2, x4)
U11_GGAGAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_GGAGAA(x1, x2, x4, x7)
U12_GGAGAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GGAGAA(x1, x2, x3, x4, x7)
PM_IN_GAGA(x1, x2, x3, x4)  =  PM_IN_GAGA(x1, x3)
U13_GAGA(x1, x2, x3, x4, x5)  =  U13_GAGA(x1, x3, x5)
MERGESORTE_IN_GA(x1, x2)  =  MERGESORTE_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
PF_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PF_IN_GGGAAAAA(x1, x2, x3)
U15_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_GGGAAAAA(x1, x2, x3, x9)
U16_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGAAAAA(x1, x2, x3, x4, x5, x9)
PN_IN_GAGAA(x1, x2, x3, x4, x5)  =  PN_IN_GAGAA(x1, x3)
U17_GAGAA(x1, x2, x3, x4, x5, x6)  =  U17_GAGAA(x1, x3, x6)
U18_GAGAA(x1, x2, x3, x4, x5, x6)  =  U18_GAGAA(x1, x2, x3, x6)
PO_IN_GAGA(x1, x2, x3, x4)  =  PO_IN_GAGA(x1, x3)
U19_GAGA(x1, x2, x3, x4, x5)  =  U19_GAGA(x1, x3, x5)
U20_GAGA(x1, x2, x3, x4, x5)  =  U20_GAGA(x1, x2, x3, x5)
MERGEG_IN_GGA(x1, x2, x3)  =  MERGEG_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
PH_IN_GGGGA(x1, x2, x3, x4, x5)  =  PH_IN_GGGGA(x1, x2, x3, x4)
U21_GGGGA(x1, x2, x3, x4, x5, x6)  =  U21_GGGGA(x1, x2, x3, x4, x6)
LEJ_IN_GG(x1, x2)  =  LEJ_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U22_GGGGA(x1, x2, x3, x4, x5, x6)  =  U22_GGGGA(x1, x2, x3, x4, x6)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)
PI_IN_GGGGA(x1, x2, x3, x4, x5)  =  PI_IN_GGGGA(x1, x2, x3, x4)
U23_GGGGA(x1, x2, x3, x4, x5, x6)  =  U23_GGGGA(x1, x2, x3, x4, x6)
GTK_IN_GG(x1, x2)  =  GTK_IN_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)
U24_GGGGA(x1, x2, x3, x4, x5, x6)  =  U24_GGGGA(x1, x2, x3, x4, x6)
U14_GAGA(x1, x2, x3, x4, x5)  =  U14_GAGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 28 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GTK_IN_GG(s(T149), s(T150)) → GTK_IN_GG(T149, T150)

The TRS R consists of the following rules:

mergesortA_in_ga([], []) → mergesortA_out_ga([], [])
mergesortA_in_ga(.(T4, []), .(T4, [])) → mergesortA_out_ga(.(T4, []), .(T4, []))
mergesortA_in_ga(.(T21, .(T22, T23)), T14) → U1_ga(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
pB_in_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14) → U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
pL_in_ggagaa(T21, T25, T38, T24, X23, T14) → U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
pM_in_gaga(T24, T39, T38, T14) → U13_gaga(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
U13_gaga(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_gaga(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U14_gaga(T24, T39, T38, T14, mergeG_out_gga(T38, T39, T14)) → pM_out_gaga(T24, T39, T38, T14)
U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_out_gaga(T24, X23, T38, T14)) → pL_out_ggagaa(T21, T25, T38, T24, X23, T14)
U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_out_ggagaa(T21, T25, X22, T24, X23, T14)) → pB_out_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14)
U1_ga(T21, T22, T23, T14, pB_out_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14)) → mergesortA_out_ga(.(T21, .(T22, T23)), T14)

The argument filtering Pi contains the following mapping:
mergesortA_in_ga(x1, x2)  =  mergesortA_in_ga(x1)
[]  =  []
mergesortA_out_ga(x1, x2)  =  mergesortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaagaaa(x1, x2, x5)
U9_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_ggaagaaa(x1, x2, x5, x9)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
U10_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_ggaagaaa(x1, x2, x3, x4, x5, x9)
pL_in_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_in_ggagaa(x1, x2, x4)
U11_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_ggagaa(x1, x2, x4, x7)
U12_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggagaa(x1, x2, x3, x4, x7)
pM_in_gaga(x1, x2, x3, x4)  =  pM_in_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5)  =  U13_gaga(x1, x3, x5)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U14_gaga(x1, x2, x3, x4, x5)  =  U14_gaga(x1, x2, x3, x5)
pM_out_gaga(x1, x2, x3, x4)  =  pM_out_gaga(x1, x2, x3, x4)
pL_out_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_out_ggagaa(x1, x2, x3, x4, x5, x6)
pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
GTK_IN_GG(x1, x2)  =  GTK_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:

GTK_IN_GG(s(T149), s(T150)) → GTK_IN_GG(T149, T150)

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:

GTK_IN_GG(s(T149), s(T150)) → GTK_IN_GG(T149, T150)

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:

• GTK_IN_GG(s(T149), s(T150)) → GTK_IN_GG(T149, T150)
  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:

LEJ_IN_GG(s(T110), s(T111)) → LEJ_IN_GG(T110, T111)

The TRS R consists of the following rules:

mergesortA_in_ga([], []) → mergesortA_out_ga([], [])
mergesortA_in_ga(.(T4, []), .(T4, [])) → mergesortA_out_ga(.(T4, []), .(T4, []))
mergesortA_in_ga(.(T21, .(T22, T23)), T14) → U1_ga(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
pB_in_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14) → U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
pL_in_ggagaa(T21, T25, T38, T24, X23, T14) → U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
pM_in_gaga(T24, T39, T38, T14) → U13_gaga(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
U13_gaga(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_gaga(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U14_gaga(T24, T39, T38, T14, mergeG_out_gga(T38, T39, T14)) → pM_out_gaga(T24, T39, T38, T14)
U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_out_gaga(T24, X23, T38, T14)) → pL_out_ggagaa(T21, T25, T38, T24, X23, T14)
U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_out_ggagaa(T21, T25, X22, T24, X23, T14)) → pB_out_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14)
U1_ga(T21, T22, T23, T14, pB_out_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14)) → mergesortA_out_ga(.(T21, .(T22, T23)), T14)

The argument filtering Pi contains the following mapping:
mergesortA_in_ga(x1, x2)  =  mergesortA_in_ga(x1)
[]  =  []
mergesortA_out_ga(x1, x2)  =  mergesortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaagaaa(x1, x2, x5)
U9_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_ggaagaaa(x1, x2, x5, x9)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
U10_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_ggaagaaa(x1, x2, x3, x4, x5, x9)
pL_in_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_in_ggagaa(x1, x2, x4)
U11_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_ggagaa(x1, x2, x4, x7)
U12_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggagaa(x1, x2, x3, x4, x7)
pM_in_gaga(x1, x2, x3, x4)  =  pM_in_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5)  =  U13_gaga(x1, x3, x5)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U14_gaga(x1, x2, x3, x4, x5)  =  U14_gaga(x1, x2, x3, x5)
pM_out_gaga(x1, x2, x3, x4)  =  pM_out_gaga(x1, x2, x3, x4)
pL_out_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_out_ggagaa(x1, x2, x3, x4, x5, x6)
pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
LEJ_IN_GG(x1, x2)  =  LEJ_IN_GG(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LEJ_IN_GG(s(T110), s(T111)) → LEJ_IN_GG(T110, T111)

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

(17) PiDPToQDPProof (EQUIVALENT 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:

LEJ_IN_GG(s(T110), s(T111)) → LEJ_IN_GG(T110, T111)

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:

• LEJ_IN_GG(s(T110), s(T111)) → LEJ_IN_GG(T110, T111)
  The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

MERGEG_IN_GGA(.(T92, T93), .(T94, T95), .(T92, T97)) → PH_IN_GGGGA(T92, T94, T93, T95, T97)
PH_IN_GGGGA(T92, T94, T93, T95, T97) → U21_GGGGA(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
U21_GGGGA(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → MERGEG_IN_GGA(T93, .(T94, T95), T97)
MERGEG_IN_GGA(.(T131, T132), .(T133, T134), .(T133, T136)) → PI_IN_GGGGA(T131, T133, T132, T134, T136)
PI_IN_GGGGA(T131, T133, T132, T134, T136) → U23_GGGGA(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
U23_GGGGA(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → MERGEG_IN_GGA(.(T131, T132), T134, T136)

The TRS R consists of the following rules:

mergesortA_in_ga([], []) → mergesortA_out_ga([], [])
mergesortA_in_ga(.(T4, []), .(T4, [])) → mergesortA_out_ga(.(T4, []), .(T4, []))
mergesortA_in_ga(.(T21, .(T22, T23)), T14) → U1_ga(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
pB_in_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14) → U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
pL_in_ggagaa(T21, T25, T38, T24, X23, T14) → U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
pM_in_gaga(T24, T39, T38, T14) → U13_gaga(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
U13_gaga(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_gaga(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U14_gaga(T24, T39, T38, T14, mergeG_out_gga(T38, T39, T14)) → pM_out_gaga(T24, T39, T38, T14)
U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_out_gaga(T24, X23, T38, T14)) → pL_out_ggagaa(T21, T25, T38, T24, X23, T14)
U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_out_ggagaa(T21, T25, X22, T24, X23, T14)) → pB_out_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14)
U1_ga(T21, T22, T23, T14, pB_out_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14)) → mergesortA_out_ga(.(T21, .(T22, T23)), T14)

The argument filtering Pi contains the following mapping:
mergesortA_in_ga(x1, x2)  =  mergesortA_in_ga(x1)
[]  =  []
mergesortA_out_ga(x1, x2)  =  mergesortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaagaaa(x1, x2, x5)
U9_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_ggaagaaa(x1, x2, x5, x9)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
U10_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_ggaagaaa(x1, x2, x3, x4, x5, x9)
pL_in_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_in_ggagaa(x1, x2, x4)
U11_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_ggagaa(x1, x2, x4, x7)
U12_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggagaa(x1, x2, x3, x4, x7)
pM_in_gaga(x1, x2, x3, x4)  =  pM_in_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5)  =  U13_gaga(x1, x3, x5)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U14_gaga(x1, x2, x3, x4, x5)  =  U14_gaga(x1, x2, x3, x5)
pM_out_gaga(x1, x2, x3, x4)  =  pM_out_gaga(x1, x2, x3, x4)
pL_out_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_out_ggagaa(x1, x2, x3, x4, x5, x6)
pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
MERGEG_IN_GGA(x1, x2, x3)  =  MERGEG_IN_GGA(x1, x2)
PH_IN_GGGGA(x1, x2, x3, x4, x5)  =  PH_IN_GGGGA(x1, x2, x3, x4)
U21_GGGGA(x1, x2, x3, x4, x5, x6)  =  U21_GGGGA(x1, x2, x3, x4, x6)
PI_IN_GGGGA(x1, x2, x3, x4, x5)  =  PI_IN_GGGGA(x1, x2, x3, x4)
U23_GGGGA(x1, x2, x3, x4, x5, x6)  =  U23_GGGGA(x1, x2, x3, x4, 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:

MERGEG_IN_GGA(.(T92, T93), .(T94, T95), .(T92, T97)) → PH_IN_GGGGA(T92, T94, T93, T95, T97)
PH_IN_GGGGA(T92, T94, T93, T95, T97) → U21_GGGGA(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
U21_GGGGA(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → MERGEG_IN_GGA(T93, .(T94, T95), T97)
MERGEG_IN_GGA(.(T131, T132), .(T133, T134), .(T133, T136)) → PI_IN_GGGGA(T131, T133, T132, T134, T136)
PI_IN_GGGGA(T131, T133, T132, T134, T136) → U23_GGGGA(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
U23_GGGGA(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → MERGEG_IN_GGA(.(T131, T132), T134, T136)

The TRS R consists of the following rules:

leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
MERGEG_IN_GGA(x1, x2, x3)  =  MERGEG_IN_GGA(x1, x2)
PH_IN_GGGGA(x1, x2, x3, x4, x5)  =  PH_IN_GGGGA(x1, x2, x3, x4)
U21_GGGGA(x1, x2, x3, x4, x5, x6)  =  U21_GGGGA(x1, x2, x3, x4, x6)
PI_IN_GGGGA(x1, x2, x3, x4, x5)  =  PI_IN_GGGGA(x1, x2, x3, x4)
U23_GGGGA(x1, x2, x3, x4, x5, x6)  =  U23_GGGGA(x1, x2, x3, x4, 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:

MERGEG_IN_GGA(.(T92, T93), .(T94, T95)) → PH_IN_GGGGA(T92, T94, T93, T95)
PH_IN_GGGGA(T92, T94, T93, T95) → U21_GGGGA(T92, T94, T93, T95, leJ_in_gg(T92, T94))
U21_GGGGA(T92, T94, T93, T95, leJ_out_gg(T92, T94)) → MERGEG_IN_GGA(T93, .(T94, T95))
MERGEG_IN_GGA(.(T131, T132), .(T133, T134)) → PI_IN_GGGGA(T131, T133, T132, T134)
PI_IN_GGGGA(T131, T133, T132, T134) → U23_GGGGA(T131, T133, T132, T134, gtK_in_gg(T131, T133))
U23_GGGGA(T131, T133, T132, T134, gtK_out_gg(T131, T133)) → MERGEG_IN_GGA(.(T131, T132), T134)

The TRS R consists of the following rules:

leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

leJ_in_gg(x0, x1)
gtK_in_gg(x0, x1)
U7_gg(x0, x1, x2)
U8_gg(x0, x1, x2)

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

(26) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MERGEG_IN_GGA(.(T92, T93), .(T94, T95)) → PH_IN_GGGGA(T92, T94, T93, T95)
MERGEG_IN_GGA(.(T131, T132), .(T133, T134)) → PI_IN_GGGGA(T131, T133, T132, T134)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(0) = 0   
POL(MERGEG_IN_GGA(x₁, x₂)) = x₁ + x₂   
POL(PH_IN_GGGGA(x₁, x₂, x₃, x₄)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄   
POL(PI_IN_GGGGA(x₁, x₂, x₃, x₄)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄   
POL(U21_GGGGA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₁ + x₂ + 2·x₃ + 2·x₄ + x₅   
POL(U23_GGGGA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₁ + x₂ + 2·x₃ + x₄ + x₅   
POL(U7_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U8_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(gtK_in_gg(x₁, x₂)) = x₁ + x₂   
POL(gtK_out_gg(x₁, x₂)) = x₁ + x₂   
POL(leJ_in_gg(x₁, x₂)) = x₁ + x₂   
POL(leJ_out_gg(x₁, x₂)) = x₁ + x₂   
POL(s(x₁)) = 2·x₁   

(27) Obligation:

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

PH_IN_GGGGA(T92, T94, T93, T95) → U21_GGGGA(T92, T94, T93, T95, leJ_in_gg(T92, T94))
U21_GGGGA(T92, T94, T93, T95, leJ_out_gg(T92, T94)) → MERGEG_IN_GGA(T93, .(T94, T95))
PI_IN_GGGGA(T131, T133, T132, T134) → U23_GGGGA(T131, T133, T132, T134, gtK_in_gg(T131, T133))
U23_GGGGA(T131, T133, T132, T134, gtK_out_gg(T131, T133)) → MERGEG_IN_GGA(.(T131, T132), T134)

The TRS R consists of the following rules:

leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

leJ_in_gg(x0, x1)
gtK_in_gg(x0, x1)
U7_gg(x0, x1, x2)
U8_gg(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 4 less nodes.

(29) TRUE

(30) Obligation:

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

SPLITC_IN_GAA(.(T36, T37), .(T36, X76), X77) → SPLITC_IN_GAA(T37, X77, X76)

The TRS R consists of the following rules:

mergesortA_in_ga([], []) → mergesortA_out_ga([], [])
mergesortA_in_ga(.(T4, []), .(T4, [])) → mergesortA_out_ga(.(T4, []), .(T4, []))
mergesortA_in_ga(.(T21, .(T22, T23)), T14) → U1_ga(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
pB_in_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14) → U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
pL_in_ggagaa(T21, T25, T38, T24, X23, T14) → U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
pM_in_gaga(T24, T39, T38, T14) → U13_gaga(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
U13_gaga(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_gaga(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U14_gaga(T24, T39, T38, T14, mergeG_out_gga(T38, T39, T14)) → pM_out_gaga(T24, T39, T38, T14)
U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_out_gaga(T24, X23, T38, T14)) → pL_out_ggagaa(T21, T25, T38, T24, X23, T14)
U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_out_ggagaa(T21, T25, X22, T24, X23, T14)) → pB_out_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14)
U1_ga(T21, T22, T23, T14, pB_out_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14)) → mergesortA_out_ga(.(T21, .(T22, T23)), T14)

The argument filtering Pi contains the following mapping:
mergesortA_in_ga(x1, x2)  =  mergesortA_in_ga(x1)
[]  =  []
mergesortA_out_ga(x1, x2)  =  mergesortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaagaaa(x1, x2, x5)
U9_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_ggaagaaa(x1, x2, x5, x9)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
U10_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_ggaagaaa(x1, x2, x3, x4, x5, x9)
pL_in_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_in_ggagaa(x1, x2, x4)
U11_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_ggagaa(x1, x2, x4, x7)
U12_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggagaa(x1, x2, x3, x4, x7)
pM_in_gaga(x1, x2, x3, x4)  =  pM_in_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5)  =  U13_gaga(x1, x3, x5)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U14_gaga(x1, x2, x3, x4, x5)  =  U14_gaga(x1, x2, x3, x5)
pM_out_gaga(x1, x2, x3, x4)  =  pM_out_gaga(x1, x2, x3, x4)
pL_out_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_out_ggagaa(x1, x2, x3, x4, x5, x6)
pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
SPLITC_IN_GAA(x1, x2, x3)  =  SPLITC_IN_GAA(x1)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

SPLITC_IN_GAA(.(T36, T37), .(T36, X76), X77) → SPLITC_IN_GAA(T37, X77, X76)

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

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

SPLITC_IN_GAA(.(T36, T37)) → SPLITC_IN_GAA(T37)

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

(35) 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:

• SPLITC_IN_GAA(.(T36, T37)) → SPLITC_IN_GAA(T37)
  The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

PF_IN_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105) → U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → PN_IN_GAGAA(T54, X103, T55, X104, X105)
PN_IN_GAGAA(T54, T56, T55, X104, X105) → U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → PO_IN_GAGA(T55, X104, T56, X105)
PO_IN_GAGA(T55, T57, T56, X105) → MERGESORTE_IN_GA(T55, T57)
MERGESORTE_IN_GA(.(T51, .(T52, T53)), X105) → PF_IN_GGGAAAAA(T51, T52, T53, X101, X102, X103, X104, X105)
PN_IN_GAGAA(T54, T56, T55, X104, X105) → MERGESORTE_IN_GA(T54, T56)

The TRS R consists of the following rules:

mergesortA_in_ga([], []) → mergesortA_out_ga([], [])
mergesortA_in_ga(.(T4, []), .(T4, [])) → mergesortA_out_ga(.(T4, []), .(T4, []))
mergesortA_in_ga(.(T21, .(T22, T23)), T14) → U1_ga(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
pB_in_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14) → U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
pL_in_ggagaa(T21, T25, T38, T24, X23, T14) → U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
pM_in_gaga(T24, T39, T38, T14) → U13_gaga(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
U13_gaga(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_gaga(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U14_gaga(T24, T39, T38, T14, mergeG_out_gga(T38, T39, T14)) → pM_out_gaga(T24, T39, T38, T14)
U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_out_gaga(T24, X23, T38, T14)) → pL_out_ggagaa(T21, T25, T38, T24, X23, T14)
U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_out_ggagaa(T21, T25, X22, T24, X23, T14)) → pB_out_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14)
U1_ga(T21, T22, T23, T14, pB_out_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14)) → mergesortA_out_ga(.(T21, .(T22, T23)), T14)

The argument filtering Pi contains the following mapping:
mergesortA_in_ga(x1, x2)  =  mergesortA_in_ga(x1)
[]  =  []
mergesortA_out_ga(x1, x2)  =  mergesortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaagaaa(x1, x2, x5)
U9_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_ggaagaaa(x1, x2, x5, x9)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
U10_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_ggaagaaa(x1, x2, x3, x4, x5, x9)
pL_in_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_in_ggagaa(x1, x2, x4)
U11_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_ggagaa(x1, x2, x4, x7)
U12_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggagaa(x1, x2, x3, x4, x7)
pM_in_gaga(x1, x2, x3, x4)  =  pM_in_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5)  =  U13_gaga(x1, x3, x5)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U14_gaga(x1, x2, x3, x4, x5)  =  U14_gaga(x1, x2, x3, x5)
pM_out_gaga(x1, x2, x3, x4)  =  pM_out_gaga(x1, x2, x3, x4)
pL_out_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_out_ggagaa(x1, x2, x3, x4, x5, x6)
pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
MERGESORTE_IN_GA(x1, x2)  =  MERGESORTE_IN_GA(x1)
PF_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PF_IN_GGGAAAAA(x1, x2, x3)
U15_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_GGGAAAAA(x1, x2, x3, x9)
PN_IN_GAGAA(x1, x2, x3, x4, x5)  =  PN_IN_GAGAA(x1, x3)
U17_GAGAA(x1, x2, x3, x4, x5, x6)  =  U17_GAGAA(x1, x3, x6)
PO_IN_GAGA(x1, x2, x3, x4)  =  PO_IN_GAGA(x1, x3)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

PF_IN_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105) → U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_GGGAAAAA(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → PN_IN_GAGAA(T54, X103, T55, X104, X105)
PN_IN_GAGAA(T54, T56, T55, X104, X105) → U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_GAGAA(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → PO_IN_GAGA(T55, X104, T56, X105)
PO_IN_GAGA(T55, T57, T56, X105) → MERGESORTE_IN_GA(T55, T57)
MERGESORTE_IN_GA(.(T51, .(T52, T53)), X105) → PF_IN_GGGAAAAA(T51, T52, T53, X101, X102, X103, X104, X105)
PN_IN_GAGAA(T54, T56, T55, X104, X105) → MERGESORTE_IN_GA(T54, T56)

The TRS R consists of the following rules:

splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
MERGESORTE_IN_GA(x1, x2)  =  MERGESORTE_IN_GA(x1)
PF_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PF_IN_GGGAAAAA(x1, x2, x3)
U15_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_GGGAAAAA(x1, x2, x3, x9)
PN_IN_GAGAA(x1, x2, x3, x4, x5)  =  PN_IN_GAGAA(x1, x3)
U17_GAGAA(x1, x2, x3, x4, x5, x6)  =  U17_GAGAA(x1, x3, x6)
PO_IN_GAGA(x1, x2, x3, x4)  =  PO_IN_GAGA(x1, x3)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

PF_IN_GGGAAAAA(T51, T52, T53) → U15_GGGAAAAA(T51, T52, T53, splitD_in_ggaa(T51, .(T52, T53)))
U15_GGGAAAAA(T51, T52, T53, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → PN_IN_GAGAA(T54, T55)
PN_IN_GAGAA(T54, T55) → U17_GAGAA(T54, T55, mergesortE_in_ga(T54))
U17_GAGAA(T54, T55, mergesortE_out_ga(T54, T56)) → PO_IN_GAGA(T55, T56)
PO_IN_GAGA(T55, T56) → MERGESORTE_IN_GA(T55)
MERGESORTE_IN_GA(.(T51, .(T52, T53))) → PF_IN_GGGAAAAA(T51, T52, T53)
PN_IN_GAGAA(T54, T55) → MERGESORTE_IN_GA(T54)

The TRS R consists of the following rules:

splitD_in_ggaa(T30, T31) → U3_ggaa(T30, T31, splitC_in_gaa(T31))
mergesortE_in_ga([]) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53))) → U4_ga(T51, T52, T53, pF_in_gggaaaaa(T51, T52, T53))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U4_ga(T51, T52, T53, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
pF_in_gggaaaaa(T51, T52, T53) → U15_gggaaaaa(T51, T52, T53, splitD_in_ggaa(T51, .(T52, T53)))
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U15_gggaaaaa(T51, T52, T53, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, pN_in_gagaa(T54, T55))
U16_gggaaaaa(T51, T52, T53, T54, T55, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
pN_in_gagaa(T54, T55) → U17_gagaa(T54, T55, mergesortE_in_ga(T54))
U17_gagaa(T54, T55, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, pO_in_gaga(T55, T56))
U18_gagaa(T54, T56, T55, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
pO_in_gaga(T55, T56) → U19_gaga(T55, T56, mergesortE_in_ga(T55))
U19_gaga(T55, T56, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, mergeG_in_gga(T56, T57))
U20_gaga(T55, T57, T56, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
mergeG_in_gga(T66, []) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95)) → U5_gga(T92, T93, T94, T95, pH_in_gggga(T92, T94, T93, T95))
mergeG_in_gga(.(T131, T132), .(T133, T134)) → U6_gga(T131, T132, T133, T134, pI_in_gggga(T131, T133, T132, T134))
U5_gga(T92, T93, T94, T95, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U6_gga(T131, T132, T133, T134, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
pH_in_gggga(T92, T94, T93, T95) → U21_gggga(T92, T94, T93, T95, leJ_in_gg(T92, T94))
pI_in_gggga(T131, T133, T132, T134) → U23_gggga(T131, T133, T132, T134, gtK_in_gg(T131, T133))
U21_gggga(T92, T94, T93, T95, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, mergeG_in_gga(T93, .(T94, T95)))
U23_gggga(T131, T133, T132, T134, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, mergeG_in_gga(.(T131, T132), T134))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U22_gggga(T92, T94, T93, T95, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U24_gggga(T131, T133, T132, T134, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

splitD_in_ggaa(x0, x1)
mergesortE_in_ga(x0)
U3_ggaa(x0, x1, x2)
U4_ga(x0, x1, x2, x3)
splitC_in_gaa(x0)
pF_in_gggaaaaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U15_gggaaaaa(x0, x1, x2, x3)
U16_gggaaaaa(x0, x1, x2, x3, x4, x5)
pN_in_gagaa(x0, x1)
U17_gagaa(x0, x1, x2)
U18_gagaa(x0, x1, x2, x3)
pO_in_gaga(x0, x1)
U19_gaga(x0, x1, x2)
U20_gaga(x0, x1, x2, x3)
mergeG_in_gga(x0, x1)
U5_gga(x0, x1, x2, x3, x4)
U6_gga(x0, x1, x2, x3, x4)
pH_in_gggga(x0, x1, x2, x3)
pI_in_gggga(x0, x1, x2, x3)
U21_gggga(x0, x1, x2, x3, x4)
U23_gggga(x0, x1, x2, x3, x4)
leJ_in_gg(x0, x1)
U22_gggga(x0, x1, x2, x3, x4)
gtK_in_gg(x0, x1)
U24_gggga(x0, x1, x2, x3, x4)
U7_gg(x0, x1, x2)
U8_gg(x0, x1, x2)

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

(42) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PF_IN_GGGAAAAA(T51, T52, T53) → U15_GGGAAAAA(T51, T52, T53, splitD_in_ggaa(T51, .(T52, T53))) at position [3] we obtained the following new rules [LPAR04]:

PF_IN_GGGAAAAA(T51, T52, T53) → U15_GGGAAAAA(T51, T52, T53, U3_ggaa(T51, .(T52, T53), splitC_in_gaa(.(T52, T53))))

(43) Obligation:

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

U15_GGGAAAAA(T51, T52, T53, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → PN_IN_GAGAA(T54, T55)
PN_IN_GAGAA(T54, T55) → U17_GAGAA(T54, T55, mergesortE_in_ga(T54))
U17_GAGAA(T54, T55, mergesortE_out_ga(T54, T56)) → PO_IN_GAGA(T55, T56)
PO_IN_GAGA(T55, T56) → MERGESORTE_IN_GA(T55)
MERGESORTE_IN_GA(.(T51, .(T52, T53))) → PF_IN_GGGAAAAA(T51, T52, T53)
PN_IN_GAGAA(T54, T55) → MERGESORTE_IN_GA(T54)
PF_IN_GGGAAAAA(T51, T52, T53) → U15_GGGAAAAA(T51, T52, T53, U3_ggaa(T51, .(T52, T53), splitC_in_gaa(.(T52, T53))))

The TRS R consists of the following rules:

splitD_in_ggaa(T30, T31) → U3_ggaa(T30, T31, splitC_in_gaa(T31))
mergesortE_in_ga([]) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53))) → U4_ga(T51, T52, T53, pF_in_gggaaaaa(T51, T52, T53))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U4_ga(T51, T52, T53, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
pF_in_gggaaaaa(T51, T52, T53) → U15_gggaaaaa(T51, T52, T53, splitD_in_ggaa(T51, .(T52, T53)))
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U15_gggaaaaa(T51, T52, T53, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, pN_in_gagaa(T54, T55))
U16_gggaaaaa(T51, T52, T53, T54, T55, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
pN_in_gagaa(T54, T55) → U17_gagaa(T54, T55, mergesortE_in_ga(T54))
U17_gagaa(T54, T55, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, pO_in_gaga(T55, T56))
U18_gagaa(T54, T56, T55, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
pO_in_gaga(T55, T56) → U19_gaga(T55, T56, mergesortE_in_ga(T55))
U19_gaga(T55, T56, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, mergeG_in_gga(T56, T57))
U20_gaga(T55, T57, T56, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
mergeG_in_gga(T66, []) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95)) → U5_gga(T92, T93, T94, T95, pH_in_gggga(T92, T94, T93, T95))
mergeG_in_gga(.(T131, T132), .(T133, T134)) → U6_gga(T131, T132, T133, T134, pI_in_gggga(T131, T133, T132, T134))
U5_gga(T92, T93, T94, T95, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U6_gga(T131, T132, T133, T134, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
pH_in_gggga(T92, T94, T93, T95) → U21_gggga(T92, T94, T93, T95, leJ_in_gg(T92, T94))
pI_in_gggga(T131, T133, T132, T134) → U23_gggga(T131, T133, T132, T134, gtK_in_gg(T131, T133))
U21_gggga(T92, T94, T93, T95, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, mergeG_in_gga(T93, .(T94, T95)))
U23_gggga(T131, T133, T132, T134, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, mergeG_in_gga(.(T131, T132), T134))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U22_gggga(T92, T94, T93, T95, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U24_gggga(T131, T133, T132, T134, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

splitD_in_ggaa(x0, x1)
mergesortE_in_ga(x0)
U3_ggaa(x0, x1, x2)
U4_ga(x0, x1, x2, x3)
splitC_in_gaa(x0)
pF_in_gggaaaaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U15_gggaaaaa(x0, x1, x2, x3)
U16_gggaaaaa(x0, x1, x2, x3, x4, x5)
pN_in_gagaa(x0, x1)
U17_gagaa(x0, x1, x2)
U18_gagaa(x0, x1, x2, x3)
pO_in_gaga(x0, x1)
U19_gaga(x0, x1, x2)
U20_gaga(x0, x1, x2, x3)
mergeG_in_gga(x0, x1)
U5_gga(x0, x1, x2, x3, x4)
U6_gga(x0, x1, x2, x3, x4)
pH_in_gggga(x0, x1, x2, x3)
pI_in_gggga(x0, x1, x2, x3)
U21_gggga(x0, x1, x2, x3, x4)
U23_gggga(x0, x1, x2, x3, x4)
leJ_in_gg(x0, x1)
U22_gggga(x0, x1, x2, x3, x4)
gtK_in_gg(x0, x1)
U24_gggga(x0, x1, x2, x3, x4)
U7_gg(x0, x1, x2)
U8_gg(x0, x1, x2)

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

(44) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PF_IN_GGGAAAAA(T51, T52, T53) → U15_GGGAAAAA(T51, T52, T53, U3_ggaa(T51, .(T52, T53), splitC_in_gaa(.(T52, T53)))) at position [3,2] we obtained the following new rules [LPAR04]:

PF_IN_GGGAAAAA(T51, T52, T53) → U15_GGGAAAAA(T51, T52, T53, U3_ggaa(T51, .(T52, T53), U2_gaa(T52, T53, splitC_in_gaa(T53))))

(45) Obligation:

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

U15_GGGAAAAA(T51, T52, T53, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → PN_IN_GAGAA(T54, T55)
PN_IN_GAGAA(T54, T55) → U17_GAGAA(T54, T55, mergesortE_in_ga(T54))
U17_GAGAA(T54, T55, mergesortE_out_ga(T54, T56)) → PO_IN_GAGA(T55, T56)
PO_IN_GAGA(T55, T56) → MERGESORTE_IN_GA(T55)
MERGESORTE_IN_GA(.(T51, .(T52, T53))) → PF_IN_GGGAAAAA(T51, T52, T53)
PN_IN_GAGAA(T54, T55) → MERGESORTE_IN_GA(T54)
PF_IN_GGGAAAAA(T51, T52, T53) → U15_GGGAAAAA(T51, T52, T53, U3_ggaa(T51, .(T52, T53), U2_gaa(T52, T53, splitC_in_gaa(T53))))

The TRS R consists of the following rules:

splitD_in_ggaa(T30, T31) → U3_ggaa(T30, T31, splitC_in_gaa(T31))
mergesortE_in_ga([]) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53))) → U4_ga(T51, T52, T53, pF_in_gggaaaaa(T51, T52, T53))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U4_ga(T51, T52, T53, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
pF_in_gggaaaaa(T51, T52, T53) → U15_gggaaaaa(T51, T52, T53, splitD_in_ggaa(T51, .(T52, T53)))
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U15_gggaaaaa(T51, T52, T53, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, pN_in_gagaa(T54, T55))
U16_gggaaaaa(T51, T52, T53, T54, T55, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
pN_in_gagaa(T54, T55) → U17_gagaa(T54, T55, mergesortE_in_ga(T54))
U17_gagaa(T54, T55, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, pO_in_gaga(T55, T56))
U18_gagaa(T54, T56, T55, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
pO_in_gaga(T55, T56) → U19_gaga(T55, T56, mergesortE_in_ga(T55))
U19_gaga(T55, T56, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, mergeG_in_gga(T56, T57))
U20_gaga(T55, T57, T56, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
mergeG_in_gga(T66, []) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95)) → U5_gga(T92, T93, T94, T95, pH_in_gggga(T92, T94, T93, T95))
mergeG_in_gga(.(T131, T132), .(T133, T134)) → U6_gga(T131, T132, T133, T134, pI_in_gggga(T131, T133, T132, T134))
U5_gga(T92, T93, T94, T95, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U6_gga(T131, T132, T133, T134, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
pH_in_gggga(T92, T94, T93, T95) → U21_gggga(T92, T94, T93, T95, leJ_in_gg(T92, T94))
pI_in_gggga(T131, T133, T132, T134) → U23_gggga(T131, T133, T132, T134, gtK_in_gg(T131, T133))
U21_gggga(T92, T94, T93, T95, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, mergeG_in_gga(T93, .(T94, T95)))
U23_gggga(T131, T133, T132, T134, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, mergeG_in_gga(.(T131, T132), T134))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U22_gggga(T92, T94, T93, T95, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U24_gggga(T131, T133, T132, T134, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

splitD_in_ggaa(x0, x1)
mergesortE_in_ga(x0)
U3_ggaa(x0, x1, x2)
U4_ga(x0, x1, x2, x3)
splitC_in_gaa(x0)
pF_in_gggaaaaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U15_gggaaaaa(x0, x1, x2, x3)
U16_gggaaaaa(x0, x1, x2, x3, x4, x5)
pN_in_gagaa(x0, x1)
U17_gagaa(x0, x1, x2)
U18_gagaa(x0, x1, x2, x3)
pO_in_gaga(x0, x1)
U19_gaga(x0, x1, x2)
U20_gaga(x0, x1, x2, x3)
mergeG_in_gga(x0, x1)
U5_gga(x0, x1, x2, x3, x4)
U6_gga(x0, x1, x2, x3, x4)
pH_in_gggga(x0, x1, x2, x3)
pI_in_gggga(x0, x1, x2, x3)
U21_gggga(x0, x1, x2, x3, x4)
U23_gggga(x0, x1, x2, x3, x4)
leJ_in_gg(x0, x1)
U22_gggga(x0, x1, x2, x3, x4)
gtK_in_gg(x0, x1)
U24_gggga(x0, x1, x2, x3, x4)
U7_gg(x0, x1, x2)
U8_gg(x0, x1, x2)

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

(46) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

MERGESORTE_IN_GA(.(T51, .(T52, T53))) → PF_IN_GGGAAAAA(T51, T52, T53)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U17_GAGAA(x₁, ..., x₃) ) = x₂

POL( mergesortE_in_ga(x₁) ) = 0

POL( [] ) = 0

POL( mergesortE_out_ga(x₁, x₂) ) = max{0, 2x₂ - 2}

POL( .(x₁, x₂) ) = x₁ + 2x₂ + 1

POL( U4_ga(x₁, ..., x₄) ) = x₄ + 2

POL( pF_in_gggaaaaa(x₁, ..., x₃) ) = 2x₁ + 1

POL( U15_GGGAAAAA(x₁, ..., x₄) ) = max{0, x₄ - 2}

POL( U2_gaa(x₁, ..., x₃) ) = max{0, x₁ + 2x₃ - 1}

POL( U3_ggaa(x₁, ..., x₃) ) = x₁ + x₃ + 1

POL( U15_gggaaaaa(x₁, ..., x₄) ) = max{0, 2x₁ + 2x₃ + x₄ - 1}

POL( splitC_in_gaa(x₁) ) = x₁ + 2

POL( splitC_out_gaa(x₁, ..., x₃) ) = x₂ + 2x₃ + 2

POL( splitD_out_ggaa(x₁, ..., x₄) ) = x₃ + x₄ + 2

POL( U16_gggaaaaa(x₁, ..., x₆) ) = max{0, x₁ + x₂ + 2x₄ - 1}

POL( pN_in_gagaa(x₁, x₂) ) = 0

POL( pN_out_gagaa(x₁, ..., x₅) ) = 2x₁ + x₂ + x₄ + 2x₅ + 1

POL( pF_out_gggaaaaa(x₁, ..., x₈) ) = x₃ + x₄ + x₅ + 2x₆ + x₇ + 2x₈ + 1

POL( U17_gagaa(x₁, ..., x₃) ) = max{0, x₂ + x₃ - 2}

POL( splitD_in_ggaa(x₁, x₂) ) = 2x₁ + x₂ + 1

POL( U18_gagaa(x₁, ..., x₄) ) = max{0, x₁ + 2x₄ - 2}

POL( pO_in_gaga(x₁, x₂) ) = x₁ + x₂

POL( pO_out_gaga(x₁, ..., x₄) ) = 2x₁ + 2x₂ + 2x₃ + 2x₄

POL( U19_gaga(x₁, ..., x₃) ) = max{0, x₁ + 2x₂ - 2}

POL( U20_gaga(x₁, ..., x₄) ) = 2x₁ + x₃ + 2

POL( mergeG_in_gga(x₁, x₂) ) = max{0, 2x₁ + x₂ - 1}

POL( mergeG_out_gga(x₁, ..., x₃) ) = max{0, x₂ + x₃ - 2}

POL( U5_gga(x₁, ..., x₅) ) = 2

POL( pH_in_gggga(x₁, ..., x₄) ) = 1

POL( U6_gga(x₁, ..., x₅) ) = max{0, x₃ + x₄ - 2}

POL( pI_in_gggga(x₁, ..., x₄) ) = x₁ + 2x₂ + x₃ + x₄

POL( pH_out_gggga(x₁, ..., x₅) ) = x₁ + 2x₂ + x₃ + 2x₄ + 2x₅ + 2

POL( U21_gggga(x₁, ..., x₅) ) = x₁ + x₂ + 2x₃ + 2x₅ + 2

POL( leJ_in_gg(x₁, x₂) ) = x₁ + x₂

POL( s(x₁) ) = 0

POL( U7_gg(x₁, ..., x₃) ) = max{0, x₂ - 2}

POL( 0 ) = 0

POL( leJ_out_gg(x₁, x₂) ) = 2x₂

POL( U22_gggga(x₁, ..., x₅) ) = 2x₁ + x₂ + 2x₃ + 2x₄ + 2

POL( pI_out_gggga(x₁, ..., x₅) ) = 2x₁ + x₂ + x₃ + x₄ + 2x₅ + 2

POL( U23_gggga(x₁, ..., x₅) ) = max{0, 2x₃ + 2x₄ + x₅ - 2}

POL( gtK_in_gg(x₁, x₂) ) = 2x₂

POL( U8_gg(x₁, ..., x₃) ) = max{0, -2}

POL( gtK_out_gg(x₁, x₂) ) = max{0, 2x₂ - 1}

POL( U24_gggga(x₁, ..., x₅) ) = max{0, x₁ + 2x₂ + 2x₃ + x₄ - 2}

POL( PN_IN_GAGAA(x₁, x₂) ) = x₁ + x₂

POL( PO_IN_GAGA(x₁, x₂) ) = x₁

POL( MERGESORTE_IN_GA(x₁) ) = x₁

POL( PF_IN_GGGAAAAA(x₁, ..., x₃) ) = x₁ + x₂ + 2x₃ + 2

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

splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)

(47) Obligation:

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

U15_GGGAAAAA(T51, T52, T53, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → PN_IN_GAGAA(T54, T55)
PN_IN_GAGAA(T54, T55) → U17_GAGAA(T54, T55, mergesortE_in_ga(T54))
U17_GAGAA(T54, T55, mergesortE_out_ga(T54, T56)) → PO_IN_GAGA(T55, T56)
PO_IN_GAGA(T55, T56) → MERGESORTE_IN_GA(T55)
PN_IN_GAGAA(T54, T55) → MERGESORTE_IN_GA(T54)
PF_IN_GGGAAAAA(T51, T52, T53) → U15_GGGAAAAA(T51, T52, T53, U3_ggaa(T51, .(T52, T53), U2_gaa(T52, T53, splitC_in_gaa(T53))))

The TRS R consists of the following rules:

splitD_in_ggaa(T30, T31) → U3_ggaa(T30, T31, splitC_in_gaa(T31))
mergesortE_in_ga([]) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53))) → U4_ga(T51, T52, T53, pF_in_gggaaaaa(T51, T52, T53))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U4_ga(T51, T52, T53, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
pF_in_gggaaaaa(T51, T52, T53) → U15_gggaaaaa(T51, T52, T53, splitD_in_ggaa(T51, .(T52, T53)))
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U15_gggaaaaa(T51, T52, T53, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, pN_in_gagaa(T54, T55))
U16_gggaaaaa(T51, T52, T53, T54, T55, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
pN_in_gagaa(T54, T55) → U17_gagaa(T54, T55, mergesortE_in_ga(T54))
U17_gagaa(T54, T55, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, pO_in_gaga(T55, T56))
U18_gagaa(T54, T56, T55, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
pO_in_gaga(T55, T56) → U19_gaga(T55, T56, mergesortE_in_ga(T55))
U19_gaga(T55, T56, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, mergeG_in_gga(T56, T57))
U20_gaga(T55, T57, T56, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
mergeG_in_gga(T66, []) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95)) → U5_gga(T92, T93, T94, T95, pH_in_gggga(T92, T94, T93, T95))
mergeG_in_gga(.(T131, T132), .(T133, T134)) → U6_gga(T131, T132, T133, T134, pI_in_gggga(T131, T133, T132, T134))
U5_gga(T92, T93, T94, T95, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U6_gga(T131, T132, T133, T134, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
pH_in_gggga(T92, T94, T93, T95) → U21_gggga(T92, T94, T93, T95, leJ_in_gg(T92, T94))
pI_in_gggga(T131, T133, T132, T134) → U23_gggga(T131, T133, T132, T134, gtK_in_gg(T131, T133))
U21_gggga(T92, T94, T93, T95, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, mergeG_in_gga(T93, .(T94, T95)))
U23_gggga(T131, T133, T132, T134, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, mergeG_in_gga(.(T131, T132), T134))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U22_gggga(T92, T94, T93, T95, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U24_gggga(T131, T133, T132, T134, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))

The set Q consists of the following terms:

splitD_in_ggaa(x0, x1)
mergesortE_in_ga(x0)
U3_ggaa(x0, x1, x2)
U4_ga(x0, x1, x2, x3)
splitC_in_gaa(x0)
pF_in_gggaaaaa(x0, x1, x2)
U2_gaa(x0, x1, x2)
U15_gggaaaaa(x0, x1, x2, x3)
U16_gggaaaaa(x0, x1, x2, x3, x4, x5)
pN_in_gagaa(x0, x1)
U17_gagaa(x0, x1, x2)
U18_gagaa(x0, x1, x2, x3)
pO_in_gaga(x0, x1)
U19_gaga(x0, x1, x2)
U20_gaga(x0, x1, x2, x3)
mergeG_in_gga(x0, x1)
U5_gga(x0, x1, x2, x3, x4)
U6_gga(x0, x1, x2, x3, x4)
pH_in_gggga(x0, x1, x2, x3)
pI_in_gggga(x0, x1, x2, x3)
U21_gggga(x0, x1, x2, x3, x4)
U23_gggga(x0, x1, x2, x3, x4)
leJ_in_gg(x0, x1)
U22_gggga(x0, x1, x2, x3, x4)
gtK_in_gg(x0, x1)
U24_gggga(x0, x1, x2, x3, x4)
U7_gg(x0, x1, x2)
U8_gg(x0, x1, x2)

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes.

(49) TRUE

(50) Obligation:

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

MERGESORTA_IN_GA(.(T21, .(T22, T23)), T14) → PB_IN_GGAAGAAA(T22, T23, X41, X40, T21, X22, X23, T14)
PB_IN_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14) → U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → PL_IN_GGAGAA(T21, T25, X22, T24, X23, T14)
PL_IN_GGAGAA(T21, T25, T38, T24, X23, T14) → MERGESORTA_IN_GA(.(T21, T25), T38)

The TRS R consists of the following rules:

mergesortA_in_ga([], []) → mergesortA_out_ga([], [])
mergesortA_in_ga(.(T4, []), .(T4, [])) → mergesortA_out_ga(.(T4, []), .(T4, []))
mergesortA_in_ga(.(T21, .(T22, T23)), T14) → U1_ga(T21, T22, T23, T14, pB_in_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14))
pB_in_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14) → U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U9_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_in_ggagaa(T21, T25, X22, T24, X23, T14))
pL_in_ggagaa(T21, T25, T38, T24, X23, T14) → U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_in_ga(.(T21, T25), T38))
U11_ggagaa(T21, T25, T38, T24, X23, T14, mergesortA_out_ga(.(T21, T25), T38)) → U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_in_gaga(T24, X23, T38, T14))
pM_in_gaga(T24, T39, T38, T14) → U13_gaga(T24, T39, T38, T14, mergesortE_in_ga(T24, T39))
mergesortE_in_ga([], []) → mergesortE_out_ga([], [])
mergesortE_in_ga(.(T44, []), .(T44, [])) → mergesortE_out_ga(.(T44, []), .(T44, []))
mergesortE_in_ga(.(T51, .(T52, T53)), X105) → U4_ga(T51, T52, T53, X105, pF_in_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105))
pF_in_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105) → U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_in_ggaa(T51, .(T52, T53), T54, T55))
U15_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, splitD_out_ggaa(T51, .(T52, T53), T54, T55)) → U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_in_gagaa(T54, X103, T55, X104, X105))
pN_in_gagaa(T54, T56, T55, X104, X105) → U17_gagaa(T54, T56, T55, X104, X105, mergesortE_in_ga(T54, T56))
U17_gagaa(T54, T56, T55, X104, X105, mergesortE_out_ga(T54, T56)) → U18_gagaa(T54, T56, T55, X104, X105, pO_in_gaga(T55, X104, T56, X105))
pO_in_gaga(T55, T57, T56, X105) → U19_gaga(T55, T57, T56, X105, mergesortE_in_ga(T55, T57))
U19_gaga(T55, T57, T56, X105, mergesortE_out_ga(T55, T57)) → U20_gaga(T55, T57, T56, X105, mergeG_in_gga(T56, T57, X105))
mergeG_in_gga(T66, [], T66) → mergeG_out_gga(T66, [], T66)
mergeG_in_gga([], T71, T71) → mergeG_out_gga([], T71, T71)
mergeG_in_gga(.(T92, T93), .(T94, T95), .(T92, T97)) → U5_gga(T92, T93, T94, T95, T97, pH_in_gggga(T92, T94, T93, T95, T97))
pH_in_gggga(T92, T94, T93, T95, T97) → U21_gggga(T92, T94, T93, T95, T97, leJ_in_gg(T92, T94))
leJ_in_gg(s(T110), s(T111)) → U7_gg(T110, T111, leJ_in_gg(T110, T111))
leJ_in_gg(0, s(T118)) → leJ_out_gg(0, s(T118))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T110, T111, leJ_out_gg(T110, T111)) → leJ_out_gg(s(T110), s(T111))
U21_gggga(T92, T94, T93, T95, T97, leJ_out_gg(T92, T94)) → U22_gggga(T92, T94, T93, T95, T97, mergeG_in_gga(T93, .(T94, T95), T97))
mergeG_in_gga(.(T131, T132), .(T133, T134), .(T133, T136)) → U6_gga(T131, T132, T133, T134, T136, pI_in_gggga(T131, T133, T132, T134, T136))
pI_in_gggga(T131, T133, T132, T134, T136) → U23_gggga(T131, T133, T132, T134, T136, gtK_in_gg(T131, T133))
gtK_in_gg(s(T149), s(T150)) → U8_gg(T149, T150, gtK_in_gg(T149, T150))
gtK_in_gg(s(T155), 0) → gtK_out_gg(s(T155), 0)
U8_gg(T149, T150, gtK_out_gg(T149, T150)) → gtK_out_gg(s(T149), s(T150))
U23_gggga(T131, T133, T132, T134, T136, gtK_out_gg(T131, T133)) → U24_gggga(T131, T133, T132, T134, T136, mergeG_in_gga(.(T131, T132), T134, T136))
U24_gggga(T131, T133, T132, T134, T136, mergeG_out_gga(.(T131, T132), T134, T136)) → pI_out_gggga(T131, T133, T132, T134, T136)
U6_gga(T131, T132, T133, T134, T136, pI_out_gggga(T131, T133, T132, T134, T136)) → mergeG_out_gga(.(T131, T132), .(T133, T134), .(T133, T136))
U22_gggga(T92, T94, T93, T95, T97, mergeG_out_gga(T93, .(T94, T95), T97)) → pH_out_gggga(T92, T94, T93, T95, T97)
U5_gga(T92, T93, T94, T95, T97, pH_out_gggga(T92, T94, T93, T95, T97)) → mergeG_out_gga(.(T92, T93), .(T94, T95), .(T92, T97))
U20_gaga(T55, T57, T56, X105, mergeG_out_gga(T56, T57, X105)) → pO_out_gaga(T55, T57, T56, X105)
U18_gagaa(T54, T56, T55, X104, X105, pO_out_gaga(T55, X104, T56, X105)) → pN_out_gagaa(T54, T56, T55, X104, X105)
U16_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105, pN_out_gagaa(T54, X103, T55, X104, X105)) → pF_out_gggaaaaa(T51, T52, T53, T54, T55, X103, X104, X105)
U4_ga(T51, T52, T53, X105, pF_out_gggaaaaa(T51, T52, T53, X101, X102, X103, X104, X105)) → mergesortE_out_ga(.(T51, .(T52, T53)), X105)
U13_gaga(T24, T39, T38, T14, mergesortE_out_ga(T24, T39)) → U14_gaga(T24, T39, T38, T14, mergeG_in_gga(T38, T39, T14))
U14_gaga(T24, T39, T38, T14, mergeG_out_gga(T38, T39, T14)) → pM_out_gaga(T24, T39, T38, T14)
U12_ggagaa(T21, T25, T38, T24, X23, T14, pM_out_gaga(T24, X23, T38, T14)) → pL_out_ggagaa(T21, T25, T38, T24, X23, T14)
U10_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14, pL_out_ggagaa(T21, T25, X22, T24, X23, T14)) → pB_out_ggaagaaa(T22, T23, T24, T25, T21, X22, X23, T14)
U1_ga(T21, T22, T23, T14, pB_out_ggaagaaa(T22, T23, X41, X40, T21, X22, X23, T14)) → mergesortA_out_ga(.(T21, .(T22, T23)), T14)

The argument filtering Pi contains the following mapping:
mergesortA_in_ga(x1, x2)  =  mergesortA_in_ga(x1)
[]  =  []
mergesortA_out_ga(x1, x2)  =  mergesortA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_ggaagaaa(x1, x2, x5)
U9_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_ggaagaaa(x1, x2, x5, x9)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
U10_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_ggaagaaa(x1, x2, x3, x4, x5, x9)
pL_in_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_in_ggagaa(x1, x2, x4)
U11_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_ggagaa(x1, x2, x4, x7)
U12_ggagaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_ggagaa(x1, x2, x3, x4, x7)
pM_in_gaga(x1, x2, x3, x4)  =  pM_in_gaga(x1, x3)
U13_gaga(x1, x2, x3, x4, x5)  =  U13_gaga(x1, x3, x5)
mergesortE_in_ga(x1, x2)  =  mergesortE_in_ga(x1)
mergesortE_out_ga(x1, x2)  =  mergesortE_out_ga(x1, x2)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
pF_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_in_gggaaaaa(x1, x2, x3)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_gggaaaaa(x1, x2, x3, x9)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_gggaaaaa(x1, x2, x3, x4, x5, x9)
pN_in_gagaa(x1, x2, x3, x4, x5)  =  pN_in_gagaa(x1, x3)
U17_gagaa(x1, x2, x3, x4, x5, x6)  =  U17_gagaa(x1, x3, x6)
U18_gagaa(x1, x2, x3, x4, x5, x6)  =  U18_gagaa(x1, x2, x3, x6)
pO_in_gaga(x1, x2, x3, x4)  =  pO_in_gaga(x1, x3)
U19_gaga(x1, x2, x3, x4, x5)  =  U19_gaga(x1, x3, x5)
U20_gaga(x1, x2, x3, x4, x5)  =  U20_gaga(x1, x2, x3, x5)
mergeG_in_gga(x1, x2, x3)  =  mergeG_in_gga(x1, x2)
mergeG_out_gga(x1, x2, x3)  =  mergeG_out_gga(x1, x2, x3)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
pH_in_gggga(x1, x2, x3, x4, x5)  =  pH_in_gggga(x1, x2, x3, x4)
U21_gggga(x1, x2, x3, x4, x5, x6)  =  U21_gggga(x1, x2, x3, x4, x6)
leJ_in_gg(x1, x2)  =  leJ_in_gg(x1, x2)
s(x1)  =  s(x1)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
0  =  0
leJ_out_gg(x1, x2)  =  leJ_out_gg(x1, x2)
U22_gggga(x1, x2, x3, x4, x5, x6)  =  U22_gggga(x1, x2, x3, x4, x6)
U6_gga(x1, x2, x3, x4, x5, x6)  =  U6_gga(x1, x2, x3, x4, x6)
pI_in_gggga(x1, x2, x3, x4, x5)  =  pI_in_gggga(x1, x2, x3, x4)
U23_gggga(x1, x2, x3, x4, x5, x6)  =  U23_gggga(x1, x2, x3, x4, x6)
gtK_in_gg(x1, x2)  =  gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2)  =  gtK_out_gg(x1, x2)
U24_gggga(x1, x2, x3, x4, x5, x6)  =  U24_gggga(x1, x2, x3, x4, x6)
pI_out_gggga(x1, x2, x3, x4, x5)  =  pI_out_gggga(x1, x2, x3, x4, x5)
pH_out_gggga(x1, x2, x3, x4, x5)  =  pH_out_gggga(x1, x2, x3, x4, x5)
pO_out_gaga(x1, x2, x3, x4)  =  pO_out_gaga(x1, x2, x3, x4)
pN_out_gagaa(x1, x2, x3, x4, x5)  =  pN_out_gagaa(x1, x2, x3, x4, x5)
pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pF_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U14_gaga(x1, x2, x3, x4, x5)  =  U14_gaga(x1, x2, x3, x5)
pM_out_gaga(x1, x2, x3, x4)  =  pM_out_gaga(x1, x2, x3, x4)
pL_out_ggagaa(x1, x2, x3, x4, x5, x6)  =  pL_out_ggagaa(x1, x2, x3, x4, x5, x6)
pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
MERGESORTA_IN_GA(x1, x2)  =  MERGESORTA_IN_GA(x1)
PB_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGAAGAAA(x1, x2, x5)
U9_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_GGAAGAAA(x1, x2, x5, x9)
PL_IN_GGAGAA(x1, x2, x3, x4, x5, x6)  =  PL_IN_GGAGAA(x1, x2, x4)

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

MERGESORTA_IN_GA(.(T21, .(T22, T23)), T14) → PB_IN_GGAAGAAA(T22, T23, X41, X40, T21, X22, X23, T14)
PB_IN_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14) → U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_in_ggaa(T22, T23, T24, T25))
U9_GGAAGAAA(T22, T23, T24, T25, T21, X22, X23, T14, splitD_out_ggaa(T22, T23, T24, T25)) → PL_IN_GGAGAA(T21, T25, X22, T24, X23, T14)
PL_IN_GGAGAA(T21, T25, T38, T24, X23, T14) → MERGESORTA_IN_GA(.(T21, T25), T38)

The TRS R consists of the following rules:

splitD_in_ggaa(T30, T31, .(T30, X58), X59) → U3_ggaa(T30, T31, X58, X59, splitC_in_gaa(T31, X59, X58))
U3_ggaa(T30, T31, X58, X59, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
splitC_in_gaa([], [], []) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37), .(T36, X76), X77) → U2_gaa(T36, T37, X76, X77, splitC_in_gaa(T37, X77, X76))
U2_gaa(T36, T37, X76, X77, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
splitD_in_ggaa(x1, x2, x3, x4)  =  splitD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
splitC_in_gaa(x1, x2, x3)  =  splitC_in_gaa(x1)
splitC_out_gaa(x1, x2, x3)  =  splitC_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x2, x5)
splitD_out_ggaa(x1, x2, x3, x4)  =  splitD_out_ggaa(x1, x2, x3, x4)
MERGESORTA_IN_GA(x1, x2)  =  MERGESORTA_IN_GA(x1)
PB_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGAAGAAA(x1, x2, x5)
U9_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U9_GGAAGAAA(x1, x2, x5, x9)
PL_IN_GGAGAA(x1, x2, x3, x4, x5, x6)  =  PL_IN_GGAGAA(x1, x2, x4)

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

MERGESORTA_IN_GA(.(T21, .(T22, T23))) → PB_IN_GGAAGAAA(T22, T23, T21)
PB_IN_GGAAGAAA(T22, T23, T21) → U9_GGAAGAAA(T22, T23, T21, splitD_in_ggaa(T22, T23))
U9_GGAAGAAA(T22, T23, T21, splitD_out_ggaa(T22, T23, T24, T25)) → PL_IN_GGAGAA(T21, T25, T24)
PL_IN_GGAGAA(T21, T25, T24) → MERGESORTA_IN_GA(.(T21, T25))

The TRS R consists of the following rules:

splitD_in_ggaa(T30, T31) → U3_ggaa(T30, T31, splitC_in_gaa(T31))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)

The set Q consists of the following terms:

splitD_in_ggaa(x0, x1)
U3_ggaa(x0, x1, x2)
splitC_in_gaa(x0)
U2_gaa(x0, x1, x2)

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

(55) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PB_IN_GGAAGAAA(T22, T23, T21) → U9_GGAAGAAA(T22, T23, T21, splitD_in_ggaa(T22, T23)) at position [3] we obtained the following new rules [LPAR04]:

PB_IN_GGAAGAAA(T22, T23, T21) → U9_GGAAGAAA(T22, T23, T21, U3_ggaa(T22, T23, splitC_in_gaa(T23)))

(56) Obligation:

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

MERGESORTA_IN_GA(.(T21, .(T22, T23))) → PB_IN_GGAAGAAA(T22, T23, T21)
U9_GGAAGAAA(T22, T23, T21, splitD_out_ggaa(T22, T23, T24, T25)) → PL_IN_GGAGAA(T21, T25, T24)
PL_IN_GGAGAA(T21, T25, T24) → MERGESORTA_IN_GA(.(T21, T25))
PB_IN_GGAAGAAA(T22, T23, T21) → U9_GGAAGAAA(T22, T23, T21, U3_ggaa(T22, T23, splitC_in_gaa(T23)))

The TRS R consists of the following rules:

splitD_in_ggaa(T30, T31) → U3_ggaa(T30, T31, splitC_in_gaa(T31))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)

The set Q consists of the following terms:

splitD_in_ggaa(x0, x1)
U3_ggaa(x0, x1, x2)
splitC_in_gaa(x0)
U2_gaa(x0, x1, x2)

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

(57) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(58) Obligation:

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

MERGESORTA_IN_GA(.(T21, .(T22, T23))) → PB_IN_GGAAGAAA(T22, T23, T21)
U9_GGAAGAAA(T22, T23, T21, splitD_out_ggaa(T22, T23, T24, T25)) → PL_IN_GGAGAA(T21, T25, T24)
PL_IN_GGAGAA(T21, T25, T24) → MERGESORTA_IN_GA(.(T21, T25))
PB_IN_GGAAGAAA(T22, T23, T21) → U9_GGAAGAAA(T22, T23, T21, U3_ggaa(T22, T23, splitC_in_gaa(T23)))

The TRS R consists of the following rules:

splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)

The set Q consists of the following terms:

splitD_in_ggaa(x0, x1)
U3_ggaa(x0, x1, x2)
splitC_in_gaa(x0)
U2_gaa(x0, x1, x2)

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

(59) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

splitD_in_ggaa(x0, x1)

(60) Obligation:

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

MERGESORTA_IN_GA(.(T21, .(T22, T23))) → PB_IN_GGAAGAAA(T22, T23, T21)
U9_GGAAGAAA(T22, T23, T21, splitD_out_ggaa(T22, T23, T24, T25)) → PL_IN_GGAGAA(T21, T25, T24)
PL_IN_GGAGAA(T21, T25, T24) → MERGESORTA_IN_GA(.(T21, T25))
PB_IN_GGAAGAAA(T22, T23, T21) → U9_GGAAGAAA(T22, T23, T21, U3_ggaa(T22, T23, splitC_in_gaa(T23)))

The TRS R consists of the following rules:

splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)

The set Q consists of the following terms:

U3_ggaa(x0, x1, x2)
splitC_in_gaa(x0)
U2_gaa(x0, x1, x2)

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

(61) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

MERGESORTA_IN_GA(.(T21, .(T22, T23))) → PB_IN_GGAAGAAA(T22, T23, T21)

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(MERGESORTA_IN_GA(x₁)) = x₁   
POL(PB_IN_GGAAGAAA(x₁, x₂, x₃)) = 1 + x₂   
POL(PL_IN_GGAGAA(x₁, x₂, x₃)) = 1 + x₂   
POL(U2_gaa(x₁, x₂, x₃)) = 1 + x₃   
POL(U3_ggaa(x₁, x₂, x₃)) = x₃   
POL(U9_GGAAGAAA(x₁, x₂, x₃, x₄)) = x₄   
POL([]) = 0   
POL(splitC_in_gaa(x₁)) = 1 + x₁   
POL(splitC_out_gaa(x₁, x₂, x₃)) = 1 + x₂ + x₃   
POL(splitD_out_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   

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

splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)

(62) Obligation:

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

U9_GGAAGAAA(T22, T23, T21, splitD_out_ggaa(T22, T23, T24, T25)) → PL_IN_GGAGAA(T21, T25, T24)
PL_IN_GGAGAA(T21, T25, T24) → MERGESORTA_IN_GA(.(T21, T25))
PB_IN_GGAAGAAA(T22, T23, T21) → U9_GGAAGAAA(T22, T23, T21, U3_ggaa(T22, T23, splitC_in_gaa(T23)))

The TRS R consists of the following rules:

splitC_in_gaa([]) → splitC_out_gaa([], [], [])
splitC_in_gaa(.(T36, T37)) → U2_gaa(T36, T37, splitC_in_gaa(T37))
U3_ggaa(T30, T31, splitC_out_gaa(T31, X59, X58)) → splitD_out_ggaa(T30, T31, .(T30, X58), X59)
U2_gaa(T36, T37, splitC_out_gaa(T37, X77, X76)) → splitC_out_gaa(.(T36, T37), .(T36, X76), X77)

The set Q consists of the following terms:

U3_ggaa(x0, x1, x2)
splitC_in_gaa(x0)
U2_gaa(x0, x1, x2)

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

(63) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(64) TRUE