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

(0) Obligation:

Clauses:

qs([], []).
qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(less(X, Y), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- part(X, Xs, Ls, Bs).
part(X1, [], [], []).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
less(0, s(X2)).
less(s(X), s(Y)) :- less(X, Y).

Query: qs(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: qs(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: qs(T1, T2), SCOPE: 1, CLAUSE: 0 | qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | qs([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nqs(T1, T2) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: qs([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9))))\n\nT6 is ground\nT7 is ground\nX12 is free\nX13 is free\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 2 | ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX12 is free\nX13 is free\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX12 is free\nX13 is free\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX12 is free\nX13 is free\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="B: ','(','(less(T22, T23), part(T22, T24, X49, X50)), ','(qs(.(T23, X49), X14), ','(qs(X50, X15), app(X14, .(T22, X15), T9))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX14 is free\nX15 is free\nX49 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="E: less(T22, T23)\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="M: ','(part(T22, T24, X49, X50), ','(qs(.(T23, X49), X14), ','(qs(X50, X15), app(X14, .(T22, X15), T9))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX14 is free\nX15 is free\nX49 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(T22, T23), SCOPE: 3, CLAUSE: 7 | less(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: less(T22, T23), SCOPE: 3, CLAUSE: 7\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: less(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: less(T38, T39)\n\nT38 is ground\nT39 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="F: part(T22, T24, X49, X50)\n\nT22 is ground\nT24 is ground\nX49 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="N: ','(qs(.(T23, T45), X14), ','(qs(T46, X15), app(X14, .(T22, X15), T9)))\n\nT22 is ground\nT23 is ground\nT45 is ground\nT46 is ground\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: part(T22, T24, X49, X50), SCOPE: 4, CLAUSE: 2 | part(T22, T24, X49, X50), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X49, X50), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX49 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: part(T22, T24, X49, X50), SCOPE: 4, CLAUSE: 2\n\nT22 is ground\nT24 is ground\nX49 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: part(T22, T24, X49, X50), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X49, X50), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX49 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="G: ','(less(T62, T63), part(T62, T64, X115, X116))\n\nT62 is ground\nT63 is ground\nT64 is ground\nX115 is free\nX116 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: less(T62, T63)\n\nT62 is ground\nT63 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: part(T62, T64, X115, X116)\n\nT62 is ground\nT64 is ground\nX115 is free\nX116 is free", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: part(T22, T24, X49, X50), SCOPE: 4, CLAUSE: 3\n\nT22 is ground\nT24 is ground\nX49 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: part(T22, T24, X49, X50), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX49 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: part(T84, T86, X161, X162)\n\nT84 is ground\nT86 is ground\nX161 is free\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
155 [label="155: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: qs(.(T23, T45), X14)\n\nT23 is ground\nT45 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="O: ','(qs(T46, X15), app(T93, .(T22, X15), T9))\n\nT22 is ground\nT46 is ground\nT93 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="H: qs(T46, X15)\n\nT46 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="K: app(T93, .(T22, T94), T9)\n\nT22 is ground\nT93 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: qs(T46, X15), SCOPE: 5, CLAUSE: 0 | qs(T46, X15), SCOPE: 5, CLAUSE: 1\n\nT46 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: qs(T46, X15), SCOPE: 5, CLAUSE: 0\n\nT46 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: qs(T46, X15), SCOPE: 5, CLAUSE: 1\n\nT46 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
175 [label="175: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
190 [label="I: ','(part(T99, T100, X188, X189), ','(qs(X188, X190), ','(qs(X189, X191), app(X190, .(T99, X191), X192))))\n\nT99 is ground\nT100 is ground\nX192 is free\nX188 is free\nX189 is free\nX190 is free\nX191 is free", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="191: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: part(T99, T100, X188, X189)\n\nT99 is ground\nT100 is ground\nX188 is free\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
193 [label="P: ','(qs(T104, X190), ','(qs(T105, X191), app(X190, .(T99, X191), X192)))\n\nT99 is ground\nT104 is ground\nT105 is ground\nX192 is free\nX190 is free\nX191 is free", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="198: qs(T104, X190)\n\nT104 is ground\nX190 is free", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="Q: ','(qs(T105, X191), app(T109, .(T99, X191), X192))\n\nT99 is ground\nT105 is ground\nT109 is ground\nX192 is free\nX191 is free", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: qs(T105, X191)\n\nT105 is ground\nX191 is free", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="J: app(T109, .(T99, T110), X192)\n\nT99 is ground\nT109 is ground\nT110 is ground\nX192 is free", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: app(T109, .(T99, T110), X192), SCOPE: 6, CLAUSE: 5 | app(T109, .(T99, T110), X192), SCOPE: 6, CLAUSE: 6\n\nT99 is ground\nT109 is ground\nT110 is ground\nX192 is free", fontsize=16, style=filled, fillcolor=lightyellow];
205 [label="205: app(T109, .(T99, T110), X192), SCOPE: 6, CLAUSE: 5\n\nT99 is ground\nT109 is ground\nT110 is ground\nX192 is free", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: app(T109, .(T99, T110), X192), SCOPE: 6, CLAUSE: 6\n\nT99 is ground\nT109 is ground\nT110 is ground\nX192 is free", fontsize=16, style=filled, fillcolor=lightyellow];
207 [label="207: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
208 [label="208: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
209 [label="209: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: app(T134, .(T135, T136), X228)\n\nT134 is ground\nT135 is ground\nT136 is ground\nX228 is free", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: app(T93, .(T22, T94), T9), SCOPE: 7, CLAUSE: 5 | app(T93, .(T22, T94), T9), SCOPE: 7, CLAUSE: 6\n\nT22 is ground\nT93 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
227 [label="227: app(T93, .(T22, T94), T9), SCOPE: 7, CLAUSE: 5\n\nT22 is ground\nT93 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="228: app(T93, .(T22, T94), T9), SCOPE: 7, CLAUSE: 6\n\nT22 is ground\nT93 is ground\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
229 [label="229: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
230 [label="230: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
231 [label="231: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: app(T166, .(T167, T168), T170)\n\nT166 is ground\nT167 is ground\nT168 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
233 [label="233: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
234 [label="234: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX12 is free\nX13 is free\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
235 [label="235: ','(part(T6, T7, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(T6, X15), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX12 is free\nX13 is free\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
236 [label="C: ','(part(T187, T189, X285, X286), ','(qs(X285, X14), ','(qs(.(T188, X286), X15), app(X14, .(T187, X15), T9))))\n\nT187 is ground\nT188 is ground\nT189 is ground\nX14 is free\nX15 is free\nX285 is free\nX286 is free", fontsize=16, style=filled, fillcolor=lightyellow];
237 [label="237: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="238: part(T187, T189, X285, X286)\n\nT187 is ground\nT189 is ground\nX285 is free\nX286 is free", fontsize=16, style=filled, fillcolor=lightyellow];
239 [label="R: ','(qs(T193, X14), ','(qs(.(T188, T194), X15), app(X14, .(T187, X15), T9)))\n\nT187 is ground\nT188 is ground\nT193 is ground\nT194 is ground\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
240 [label="240: qs(T193, X14)\n\nT193 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
241 [label="241: ','(qs(.(T188, T194), X15), app(T198, .(T187, X15), T9))\n\nT187 is ground\nT188 is ground\nT194 is ground\nT198 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
242 [label="D: ','(qs([], X14), ','(qs([], X15), app(X14, .(T201, X15), T9)))\n\nT201 is ground\nX14 is free\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
243 [label="243: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
244 [label="L: qs([], X14)\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
245 [label="S: ','(qs([], X15), app(T206, .(T201, X15), T9))\n\nT201 is ground\nT206 is ground\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
246 [label="246: qs([], X14), SCOPE: 8, CLAUSE: 0 | qs([], X14), SCOPE: 8, CLAUSE: 1\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
247 [label="247: qs([], X14), SCOPE: 8, CLAUSE: 0\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
248 [label="248: qs([], X14), SCOPE: 8, CLAUSE: 1\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
249 [label="249: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
250 [label="250: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
251 [label="251: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
252 [label="252: qs([], X15)\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
253 [label="253: app(T206, .(T201, T211), T9)\n\nT201 is ground\nT206 is ground\nT211 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
254 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 254 [arrowhead = none ];
254 -> 5;

255 [label="EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 255 [arrowhead = none ];
255 -> 10;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 256 [arrowhead = none ];
256 -> 14;

257 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 257 [arrowhead = none ];
257 -> 17;

258 [label="EVAL with clause\nqs(.(X9, X10), X11) :- ','(part(X9, X10, X12, X13), ','(qs(X12, X14), ','(qs(X13, X15), app(X14, .(X9, X15), X11)))).\nand substitutionX9 -> T6,\nX10 -> T7,\nT1 -> .(T6, T7),\nT2 -> T9,\nX11 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 258 [arrowhead = none ];
258 -> 23;

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

260 [label="BACKTRACK\nfor clause: qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 260 [arrowhead = none ];
260 -> 19;

261 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 261 [arrowhead = none ];
261 -> 27;

262 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 262 [arrowhead = none ];
262 -> 30;

263 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 263 [arrowhead = none ];
263 -> 31;

264 [label="EVAL with clause\npart(X44, .(X45, X46), .(X45, X47), X48) :- ','(less(X44, X45), part(X44, X46, X47, X48)).\nand substitutionT6 -> T22,\nX44 -> T22,\nX45 -> T23,\nX46 -> T24,\nT7 -> .(T23, T24),\nX47 -> X49,\nX12 -> .(T23, X49),\nX13 -> X50,\nX48 -> X50", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 264 [arrowhead = none ];
264 -> 40;

265 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 265 [arrowhead = none ];
265 -> 43;

266 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 266 [arrowhead = none ];
266 -> 234;

267 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 267 [arrowhead = none ];
267 -> 235;

268 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
40 -> 268 [arrowhead = none ];
268 -> 49;

269 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nT23 is ground", fontsize=14, style = filled, fillcolor = violet];
40 -> 269 [arrowhead = none ];
269 -> 50;

270 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 270 [arrowhead = none ];
270 -> 51;

271 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 271 [arrowhead = none ];
271 -> 66;

272 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nT24 is ground\nT45 is ground\nT46 is ground\nreplacements:X49 -> T45,\nX50 -> T46", fontsize=14, style = filled, fillcolor = violet];
50 -> 272 [arrowhead = none ];
272 -> 67;

273 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 273 [arrowhead = none ];
273 -> 52;

274 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 274 [arrowhead = none ];
274 -> 53;

275 [label="EVAL with clause\nless(0, s(X59)).\nand substitutionT22 -> 0,\nX59 -> T33,\nT23 -> s(T33)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 275 [arrowhead = none ];
275 -> 58;

276 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 276 [arrowhead = none ];
276 -> 59;

277 [label="EVAL with clause\nless(s(X64), s(X65)) :- less(X64, X65).\nand substitutionX64 -> T38,\nT22 -> s(T38),\nX65 -> T39,\nT23 -> s(T39)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 277 [arrowhead = none ];
277 -> 61;

278 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 278 [arrowhead = none ];
278 -> 62;

279 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 279 [arrowhead = none ];
279 -> 60;

280 [label="INSTANCE with matching:\nT22 -> T38\nT23 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
61 -> 280 [arrowhead = none , style=dashed];
280 -> 49[style=dashed];

281 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 281 [arrowhead = none ];
281 -> 68;

282 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
67 -> 282 [arrowhead = none ];
282 -> 160;

283 [label="SPLIT 2\nnew knowledge:\nT23 is ground\nT45 is ground\nT93 is ground\nreplacements:X14 -> T93", fontsize=14, style = filled, fillcolor = violet];
67 -> 283 [arrowhead = none ];
283 -> 161;

284 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 284 [arrowhead = none ];
284 -> 73;

285 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
68 -> 285 [arrowhead = none ];
285 -> 74;

286 [label="EVAL with clause\npart(X110, .(X111, X112), .(X111, X113), X114) :- ','(less(X110, X111), part(X110, X112, X113, X114)).\nand substitutionT22 -> T62,\nX110 -> T62,\nX111 -> T63,\nX112 -> T64,\nT24 -> .(T63, T64),\nX113 -> X115,\nX49 -> .(T63, X115),\nX50 -> X116,\nX114 -> X116", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 286 [arrowhead = none ];
286 -> 89;

287 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 287 [arrowhead = none ];
287 -> 94;

288 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 288 [arrowhead = none ];
288 -> 127;

289 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 289 [arrowhead = none ];
289 -> 128;

290 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
89 -> 290 [arrowhead = none ];
290 -> 95;

291 [label="SPLIT 2\nnew knowledge:\nT62 is ground\nT63 is ground", fontsize=14, style = filled, fillcolor = violet];
89 -> 291 [arrowhead = none ];
291 -> 96;

292 [label="INSTANCE with matching:\nT22 -> T62\nT23 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
95 -> 292 [arrowhead = none , style=dashed];
292 -> 49[style=dashed];

293 [label="INSTANCE with matching:\nT22 -> T62\nT24 -> T64\nX49 -> X115\nX50 -> X116", fontsize=14, style = filled, fillcolor = lightgrey];
96 -> 293 [arrowhead = none , style=dashed];
293 -> 66[style=dashed];

294 [label="EVAL with clause\npart(X156, .(X157, X158), X159, .(X157, X160)) :- part(X156, X158, X159, X160).\nand substitutionT22 -> T84,\nX156 -> T84,\nX157 -> T85,\nX158 -> T86,\nT24 -> .(T85, T86),\nX49 -> X161,\nX159 -> X161,\nX160 -> X162,\nX50 -> .(T85, X162)", fontsize=14, style = filled, fillcolor = lightblue];
127 -> 294 [arrowhead = none ];
294 -> 139;

295 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
127 -> 295 [arrowhead = none ];
295 -> 140;

296 [label="EVAL with clause\npart(X172, [], [], []).\nand substitutionT22 -> T92,\nX172 -> T92,\nT24 -> [],\nX49 -> [],\nX50 -> []", fontsize=14, style = filled, fillcolor = lightblue];
128 -> 296 [arrowhead = none ];
296 -> 153;

297 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 297 [arrowhead = none ];
297 -> 154;

298 [label="INSTANCE with matching:\nT22 -> T84\nT24 -> T86\nX49 -> X161\nX50 -> X162", fontsize=14, style = filled, fillcolor = lightgrey];
139 -> 298 [arrowhead = none , style=dashed];
298 -> 66[style=dashed];

299 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
153 -> 299 [arrowhead = none ];
299 -> 155;

300 [label="INSTANCE with matching:\nT1 -> .(T23, T45)\nT2 -> X14", fontsize=14, style = filled, fillcolor = lightgrey];
160 -> 300 [arrowhead = none , style=dashed];
300 -> 3[style=dashed];

301 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
161 -> 301 [arrowhead = none ];
301 -> 170;

302 [label="SPLIT 2\nnew knowledge:\nT46 is ground\nT94 is ground\nreplacements:X15 -> T94", fontsize=14, style = filled, fillcolor = violet];
161 -> 302 [arrowhead = none ];
302 -> 171;

303 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
170 -> 303 [arrowhead = none ];
303 -> 172;

304 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
171 -> 304 [arrowhead = none ];
304 -> 226;

305 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
172 -> 305 [arrowhead = none ];
305 -> 173;

306 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
172 -> 306 [arrowhead = none ];
306 -> 174;

307 [label="EVAL with clause\nqs([], []).\nand substitutionT46 -> [],\nX15 -> []", fontsize=14, style = filled, fillcolor = lightblue];
173 -> 307 [arrowhead = none ];
307 -> 175;

308 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
173 -> 308 [arrowhead = none ];
308 -> 176;

309 [label="EVAL with clause\nqs(.(X185, X186), X187) :- ','(part(X185, X186, X188, X189), ','(qs(X188, X190), ','(qs(X189, X191), app(X190, .(X185, X191), X187)))).\nand substitutionX185 -> T99,\nX186 -> T100,\nT46 -> .(T99, T100),\nX15 -> X192,\nX187 -> X192", fontsize=14, style = filled, fillcolor = lightblue];
174 -> 309 [arrowhead = none ];
309 -> 190;

310 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
174 -> 310 [arrowhead = none ];
310 -> 191;

311 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
175 -> 311 [arrowhead = none ];
311 -> 177;

312 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
190 -> 312 [arrowhead = none ];
312 -> 192;

313 [label="SPLIT 2\nnew knowledge:\nT99 is ground\nT100 is ground\nT104 is ground\nT105 is ground\nreplacements:X188 -> T104,\nX189 -> T105", fontsize=14, style = filled, fillcolor = violet];
190 -> 313 [arrowhead = none ];
313 -> 193;

314 [label="INSTANCE with matching:\nT22 -> T99\nT24 -> T100\nX49 -> X188\nX50 -> X189", fontsize=14, style = filled, fillcolor = lightgrey];
192 -> 314 [arrowhead = none , style=dashed];
314 -> 66[style=dashed];

315 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
193 -> 315 [arrowhead = none ];
315 -> 198;

316 [label="SPLIT 2\nnew knowledge:\nT104 is ground\nT109 is ground\nreplacements:X190 -> T109", fontsize=14, style = filled, fillcolor = violet];
193 -> 316 [arrowhead = none ];
316 -> 199;

317 [label="INSTANCE with matching:\nT46 -> T104\nX15 -> X190", fontsize=14, style = filled, fillcolor = lightgrey];
198 -> 317 [arrowhead = none , style=dashed];
317 -> 170[style=dashed];

318 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
199 -> 318 [arrowhead = none ];
318 -> 202;

319 [label="SPLIT 2\nnew knowledge:\nT105 is ground\nT110 is ground\nreplacements:X191 -> T110", fontsize=14, style = filled, fillcolor = violet];
199 -> 319 [arrowhead = none ];
319 -> 203;

320 [label="INSTANCE with matching:\nT46 -> T105\nX15 -> X191", fontsize=14, style = filled, fillcolor = lightgrey];
202 -> 320 [arrowhead = none , style=dashed];
320 -> 170[style=dashed];

321 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
203 -> 321 [arrowhead = none ];
321 -> 204;

322 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
204 -> 322 [arrowhead = none ];
322 -> 205;

323 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
204 -> 323 [arrowhead = none ];
323 -> 206;

324 [label="EVAL with clause\napp([], X213, X213).\nand substitutionT109 -> [],\nT99 -> T123,\nT110 -> T124,\nX213 -> .(T123, T124),\nX192 -> .(T123, T124)", fontsize=14, style = filled, fillcolor = lightblue];
205 -> 324 [arrowhead = none ];
324 -> 207;

325 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
205 -> 325 [arrowhead = none ];
325 -> 208;

326 [label="EVAL with clause\napp(.(X224, X225), X226, .(X224, X227)) :- app(X225, X226, X227).\nand substitutionX224 -> T133,\nX225 -> T134,\nT109 -> .(T133, T134),\nT99 -> T135,\nT110 -> T136,\nX226 -> .(T135, T136),\nX227 -> X228,\nX192 -> .(T133, X228)", fontsize=14, style = filled, fillcolor = lightblue];
206 -> 326 [arrowhead = none ];
326 -> 222;

327 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
206 -> 327 [arrowhead = none ];
327 -> 223;

328 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
207 -> 328 [arrowhead = none ];
328 -> 209;

329 [label="INSTANCE with matching:\nT109 -> T134\nT99 -> T135\nT110 -> T136\nX192 -> X228", fontsize=14, style = filled, fillcolor = lightgrey];
222 -> 329 [arrowhead = none , style=dashed];
329 -> 203[style=dashed];

330 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
226 -> 330 [arrowhead = none ];
330 -> 227;

331 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
226 -> 331 [arrowhead = none ];
331 -> 228;

332 [label="EVAL with clause\napp([], X237, X237).\nand substitutionT93 -> [],\nT22 -> T153,\nT94 -> T154,\nX237 -> .(T153, T154),\nT9 -> .(T153, T154)", fontsize=14, style = filled, fillcolor = lightblue];
227 -> 332 [arrowhead = none ];
332 -> 229;

333 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
227 -> 333 [arrowhead = none ];
333 -> 230;

334 [label="EVAL with clause\napp(.(X246, X247), X248, .(X246, X249)) :- app(X247, X248, X249).\nand substitutionX246 -> T165,\nX247 -> T166,\nT93 -> .(T165, T166),\nT22 -> T167,\nT94 -> T168,\nX248 -> .(T167, T168),\nX249 -> T170,\nT9 -> .(T165, T170),\nT169 -> T170", fontsize=14, style = filled, fillcolor = lightblue];
228 -> 334 [arrowhead = none ];
334 -> 232;

335 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
228 -> 335 [arrowhead = none ];
335 -> 233;

336 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
229 -> 336 [arrowhead = none ];
336 -> 231;

337 [label="INSTANCE with matching:\nT93 -> T166\nT22 -> T167\nT94 -> T168\nT9 -> T170", fontsize=14, style = filled, fillcolor = lightgrey];
232 -> 337 [arrowhead = none , style=dashed];
337 -> 171[style=dashed];

338 [label="EVAL with clause\npart(X280, .(X281, X282), X283, .(X281, X284)) :- part(X280, X282, X283, X284).\nand substitutionT6 -> T187,\nX280 -> T187,\nX281 -> T188,\nX282 -> T189,\nT7 -> .(T188, T189),\nX12 -> X285,\nX283 -> X285,\nX284 -> X286,\nX13 -> .(T188, X286)", fontsize=14, style = filled, fillcolor = lightblue];
234 -> 338 [arrowhead = none ];
338 -> 236;

339 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
234 -> 339 [arrowhead = none ];
339 -> 237;

340 [label="EVAL with clause\npart(X303, [], [], []).\nand substitutionT6 -> T201,\nX303 -> T201,\nT7 -> [],\nX12 -> [],\nX13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
235 -> 340 [arrowhead = none ];
340 -> 242;

341 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
235 -> 341 [arrowhead = none ];
341 -> 243;

342 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
236 -> 342 [arrowhead = none ];
342 -> 238;

343 [label="SPLIT 2\nnew knowledge:\nT187 is ground\nT189 is ground\nT193 is ground\nT194 is ground\nreplacements:X285 -> T193,\nX286 -> T194", fontsize=14, style = filled, fillcolor = violet];
236 -> 343 [arrowhead = none ];
343 -> 239;

344 [label="INSTANCE with matching:\nT22 -> T187\nT24 -> T189\nX49 -> X285\nX50 -> X286", fontsize=14, style = filled, fillcolor = lightgrey];
238 -> 344 [arrowhead = none , style=dashed];
344 -> 66[style=dashed];

345 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
239 -> 345 [arrowhead = none ];
345 -> 240;

346 [label="SPLIT 2\nnew knowledge:\nT193 is ground\nT198 is ground\nreplacements:X14 -> T198", fontsize=14, style = filled, fillcolor = violet];
239 -> 346 [arrowhead = none ];
346 -> 241;

347 [label="INSTANCE with matching:\nT46 -> T193\nX15 -> X14", fontsize=14, style = filled, fillcolor = lightgrey];
240 -> 347 [arrowhead = none , style=dashed];
347 -> 170[style=dashed];

348 [label="INSTANCE with matching:\nT46 -> .(T188, T194)\nT93 -> T198\nT22 -> T187", fontsize=14, style = filled, fillcolor = lightgrey];
241 -> 348 [arrowhead = none , style=dashed];
348 -> 161[style=dashed];

349 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
242 -> 349 [arrowhead = none ];
349 -> 244;

350 [label="SPLIT 2\nnew knowledge:\nT206 is ground\nreplacements:X14 -> T206", fontsize=14, style = filled, fillcolor = violet];
242 -> 350 [arrowhead = none ];
350 -> 245;

351 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
244 -> 351 [arrowhead = none ];
351 -> 246;

352 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
245 -> 352 [arrowhead = none ];
352 -> 252;

353 [label="SPLIT 2\nnew knowledge:\nT211 is ground\nreplacements:X15 -> T211", fontsize=14, style = filled, fillcolor = violet];
245 -> 353 [arrowhead = none ];
353 -> 253;

354 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
246 -> 354 [arrowhead = none ];
354 -> 247;

355 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
246 -> 355 [arrowhead = none ];
355 -> 248;

356 [label="ONLY EVAL with clause\nqs([], []).\nand substitutionX14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
247 -> 356 [arrowhead = none ];
356 -> 249;

357 [label="BACKTRACK\nfor clause: qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
248 -> 357 [arrowhead = none ];
357 -> 251;

358 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
249 -> 358 [arrowhead = none ];
358 -> 250;

359 [label="INSTANCE with matching:\nX14 -> X15", fontsize=14, style = filled, fillcolor = lightgrey];
252 -> 359 [arrowhead = none , style=dashed];
359 -> 244[style=dashed];

360 [label="INSTANCE with matching:\nT93 -> T206\nT22 -> T201\nT94 -> T211", fontsize=14, style = filled, fillcolor = lightgrey];
253 -> 360 [arrowhead = none , style=dashed];
360 -> 171[style=dashed];

}

(2) Obligation:

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

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9) → U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
pM_in_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9) → U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
pN_in_ggagaga(T23, T45, T93, T46, X15, T22, T9) → U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
qsA_in_ga(.(T187, .(T188, T189)), T9) → U2_ga(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
pC_in_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9) → U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
pR_in_gaggaga(T193, T198, T188, T194, X15, T187, T9) → U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
pO_in_gagga(T46, T94, T93, T22, T9) → U18_gagga(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
U18_gagga(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_gagga(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
appK_in_ggga([], T153, T154, .(T153, T154)) → appK_out_ggga([], T153, T154, .(T153, T154))
appK_in_ggga(.(T165, T166), T167, T168, .(T165, T170)) → U9_ggga(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
U9_ggga(T165, T166, T167, T168, T170, appK_out_ggga(T166, T167, T168, T170)) → appK_out_ggga(.(T165, T166), T167, T168, .(T165, T170))
U19_gagga(T46, T94, T93, T22, T9, appK_out_ggga(T93, T22, T94, T9)) → pO_out_gagga(T46, T94, T93, T22, T9)
U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_out_gagga(.(T188, T194), X15, T198, T187, T9)) → pR_out_gaggaga(T193, T198, T188, T194, X15, T187, T9)
U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_out_gaggaga(T193, X14, T188, T194, X15, T187, T9)) → pC_out_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9)
U2_ga(T187, T188, T189, T9, pC_out_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9)) → qsA_out_ga(.(T187, .(T188, T189)), T9)
qsA_in_ga(.(T201, []), T9) → U3_ga(T201, T9, pD_in_aaga(X14, X15, T201, T9))
pD_in_aaga(T206, X15, T201, T9) → U30_aaga(T206, X15, T201, T9, qsL_in_a(T206))
qsL_in_a([]) → qsL_out_a([])
U30_aaga(T206, X15, T201, T9, qsL_out_a(T206)) → U31_aaga(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
pS_in_agga(T211, T206, T201, T9) → U32_agga(T211, T206, T201, T9, qsL_in_a(T211))
U32_agga(T211, T206, T201, T9, qsL_out_a(T211)) → U33_agga(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U33_agga(T211, T206, T201, T9, appK_out_ggga(T206, T201, T211, T9)) → pS_out_agga(T211, T206, T201, T9)
U31_aaga(T206, X15, T201, T9, pS_out_agga(X15, T206, T201, T9)) → pD_out_aaga(T206, X15, T201, T9)
U3_ga(T201, T9, pD_out_aaga(X14, X15, T201, T9)) → qsA_out_ga(.(T201, []), T9)
U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_out_gagga(T46, X15, T93, T22, T9)) → pN_out_ggagaga(T23, T45, T93, T46, X15, T22, T9)
U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_out_ggagaga(T23, T45, X14, T46, X15, T22, T9)) → pM_out_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9)
U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_out_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9)) → pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)) → qsA_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2)  =  qsA_in_ga(x1)
[]  =  []
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_gggaaaaa(x1, x2, x3)
U10_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_gggaaaaa(x1, x2, x3, x9)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
pM_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_in_ggaagaaa(x1, x2, x5)
U12_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_ggaagaaa(x1, x2, x5, x9)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_ggaagaaa(x1, x2, x3, x4, x5, x9)
pN_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_in_ggagaga(x1, x2, x4, x6)
U14_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_ggaaagaa(x1, x2, x6)
U26_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_ggaaagaa(x1, x2, x6, x9)
U27_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_ggaaagaa(x1, x2, x3, x4, x6, x9)
pR_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_in_gaggaga(x1, x3, x4, x6)
U28_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_gaggaga(x1, x3, x4, x6, x8)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U29_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_gaggaga(x1, x2, x3, x4, x6, x8)
pO_in_gagga(x1, x2, x3, x4, x5)  =  pO_in_gagga(x1, x3, x4)
U18_gagga(x1, x2, x3, x4, x5, x6)  =  U18_gagga(x1, x3, x4, x6)
U19_gagga(x1, x2, x3, x4, x5, x6)  =  U19_gagga(x1, x2, x3, x4, x6)
appK_in_ggga(x1, x2, x3, x4)  =  appK_in_ggga(x1, x2, x3)
appK_out_ggga(x1, x2, x3, x4)  =  appK_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
pO_out_gagga(x1, x2, x3, x4, x5)  =  pO_out_gagga(x1, x2, x3, x4, x5)
pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U30_aaga(x1, x2, x3, x4, x5)  =  U30_aaga(x3, x5)
qsL_in_a(x1)  =  qsL_in_a
qsL_out_a(x1)  =  qsL_out_a(x1)
U31_aaga(x1, x2, x3, x4, x5)  =  U31_aaga(x1, x3, x5)
pS_in_agga(x1, x2, x3, x4)  =  pS_in_agga(x2, x3)
U32_agga(x1, x2, x3, x4, x5)  =  U32_agga(x2, x3, x5)
U33_agga(x1, x2, x3, x4, x5)  =  U33_agga(x1, x2, x3, x5)
pS_out_agga(x1, x2, x3, x4)  =  pS_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x3, x4, x6, x8)
pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_gggaaaaa(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:

QSA_IN_GA(.(T22, .(T23, T24)), T9) → U1_GA(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
QSA_IN_GA(.(T22, .(T23, T24)), T9) → PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9)
PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9) → U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9) → LESSE_IN_GG(T22, T23)
LESSE_IN_GG(s(T38), s(T39)) → U4_GG(T38, T39, lessE_in_gg(T38, T39))
LESSE_IN_GG(s(T38), s(T39)) → LESSE_IN_GG(T38, T39)
U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → PM_IN_GGAAGAAA(T22, T24, X49, X50, T23, X14, X15, T9)
PM_IN_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9) → U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
PM_IN_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9) → PARTF_IN_GGAA(T22, T24, T45, T46)
PARTF_IN_GGAA(T62, .(T63, T64), .(T63, X115), X116) → U5_GGAA(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
PARTF_IN_GGAA(T62, .(T63, T64), .(T63, X115), X116) → PG_IN_GGGAA(T62, T63, T64, X115, X116)
PG_IN_GGGAA(T62, T63, T64, X115, X116) → U16_GGGAA(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
PG_IN_GGGAA(T62, T63, T64, X115, X116) → LESSE_IN_GG(T62, T63)
U16_GGGAA(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_GGGAA(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
U16_GGGAA(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → PARTF_IN_GGAA(T62, T64, X115, X116)
PARTF_IN_GGAA(T84, .(T85, T86), X161, .(T85, X162)) → U6_GGAA(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
PARTF_IN_GGAA(T84, .(T85, T86), X161, .(T85, X162)) → PARTF_IN_GGAA(T84, T86, X161, X162)
U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → PN_IN_GGAGAGA(T23, T45, X14, T46, X15, T22, T9)
PN_IN_GGAGAGA(T23, T45, T93, T46, X15, T22, T9) → U14_GGAGAGA(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
PN_IN_GGAGAGA(T23, T45, T93, T46, X15, T22, T9) → QSA_IN_GA(.(T23, T45), T93)
QSA_IN_GA(.(T187, .(T188, T189)), T9) → U2_GA(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
QSA_IN_GA(.(T187, .(T188, T189)), T9) → PC_IN_GGAAAGAA(T187, T189, X285, X286, X14, T188, X15, T9)
PC_IN_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9) → U26_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
PC_IN_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9) → PARTF_IN_GGAA(T187, T189, T193, T194)
U26_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
U26_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → PR_IN_GAGGAGA(T193, X14, T188, T194, X15, T187, T9)
PR_IN_GAGGAGA(T193, T198, T188, T194, X15, T187, T9) → U28_GAGGAGA(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
PR_IN_GAGGAGA(T193, T198, T188, T194, X15, T187, T9) → QSH_IN_GA(T193, T198)
QSH_IN_GA(.(T99, T100), X192) → U7_GA(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
QSH_IN_GA(.(T99, T100), X192) → PI_IN_GGAAAAA(T99, T100, X188, X189, X190, X191, X192)
PI_IN_GGAAAAA(T99, T100, T104, T105, X190, X191, X192) → U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
PI_IN_GGAAAAA(T99, T100, T104, T105, X190, X191, X192) → PARTF_IN_GGAA(T99, T100, T104, T105)
U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → PP_IN_GAGAGA(T104, X190, T105, X191, T99, X192)
PP_IN_GAGAGA(T104, T109, T105, X191, T99, X192) → U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
PP_IN_GAGAGA(T104, T109, T105, X191, T99, X192) → QSH_IN_GA(T104, T109)
U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_GAGAGA(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → PQ_IN_GAGGA(T105, X191, T109, T99, X192)
PQ_IN_GAGGA(T105, T110, T109, T99, X192) → U24_GAGGA(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
PQ_IN_GAGGA(T105, T110, T109, T99, X192) → QSH_IN_GA(T105, T110)
U24_GAGGA(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_GAGGA(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
U24_GAGGA(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → APPJ_IN_GGGA(T109, T99, T110, X192)
APPJ_IN_GGGA(.(T133, T134), T135, T136, .(T133, X228)) → U8_GGGA(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
APPJ_IN_GGGA(.(T133, T134), T135, T136, .(T133, X228)) → APPJ_IN_GGGA(T134, T135, T136, X228)
U28_GAGGAGA(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_GAGGAGA(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
U28_GAGGAGA(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → PO_IN_GAGGA(.(T188, T194), X15, T198, T187, T9)
PO_IN_GAGGA(T46, T94, T93, T22, T9) → U18_GAGGA(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
PO_IN_GAGGA(T46, T94, T93, T22, T9) → QSH_IN_GA(T46, T94)
U18_GAGGA(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_GAGGA(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
U18_GAGGA(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → APPK_IN_GGGA(T93, T22, T94, T9)
APPK_IN_GGGA(.(T165, T166), T167, T168, .(T165, T170)) → U9_GGGA(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
APPK_IN_GGGA(.(T165, T166), T167, T168, .(T165, T170)) → APPK_IN_GGGA(T166, T167, T168, T170)
QSA_IN_GA(.(T201, []), T9) → U3_GA(T201, T9, pD_in_aaga(X14, X15, T201, T9))
QSA_IN_GA(.(T201, []), T9) → PD_IN_AAGA(X14, X15, T201, T9)
PD_IN_AAGA(T206, X15, T201, T9) → U30_AAGA(T206, X15, T201, T9, qsL_in_a(T206))
PD_IN_AAGA(T206, X15, T201, T9) → QSL_IN_A(T206)
U30_AAGA(T206, X15, T201, T9, qsL_out_a(T206)) → U31_AAGA(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
U30_AAGA(T206, X15, T201, T9, qsL_out_a(T206)) → PS_IN_AGGA(X15, T206, T201, T9)
PS_IN_AGGA(T211, T206, T201, T9) → U32_AGGA(T211, T206, T201, T9, qsL_in_a(T211))
PS_IN_AGGA(T211, T206, T201, T9) → QSL_IN_A(T211)
U32_AGGA(T211, T206, T201, T9, qsL_out_a(T211)) → U33_AGGA(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U32_AGGA(T211, T206, T201, T9, qsL_out_a(T211)) → APPK_IN_GGGA(T206, T201, T211, T9)
U14_GGAGAGA(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_GGAGAGA(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U14_GGAGAGA(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → PO_IN_GAGGA(T46, X15, T93, T22, T9)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9) → U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
pM_in_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9) → U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
pN_in_ggagaga(T23, T45, T93, T46, X15, T22, T9) → U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
qsA_in_ga(.(T187, .(T188, T189)), T9) → U2_ga(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
pC_in_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9) → U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
pR_in_gaggaga(T193, T198, T188, T194, X15, T187, T9) → U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
pO_in_gagga(T46, T94, T93, T22, T9) → U18_gagga(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
U18_gagga(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_gagga(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
appK_in_ggga([], T153, T154, .(T153, T154)) → appK_out_ggga([], T153, T154, .(T153, T154))
appK_in_ggga(.(T165, T166), T167, T168, .(T165, T170)) → U9_ggga(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
U9_ggga(T165, T166, T167, T168, T170, appK_out_ggga(T166, T167, T168, T170)) → appK_out_ggga(.(T165, T166), T167, T168, .(T165, T170))
U19_gagga(T46, T94, T93, T22, T9, appK_out_ggga(T93, T22, T94, T9)) → pO_out_gagga(T46, T94, T93, T22, T9)
U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_out_gagga(.(T188, T194), X15, T198, T187, T9)) → pR_out_gaggaga(T193, T198, T188, T194, X15, T187, T9)
U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_out_gaggaga(T193, X14, T188, T194, X15, T187, T9)) → pC_out_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9)
U2_ga(T187, T188, T189, T9, pC_out_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9)) → qsA_out_ga(.(T187, .(T188, T189)), T9)
qsA_in_ga(.(T201, []), T9) → U3_ga(T201, T9, pD_in_aaga(X14, X15, T201, T9))
pD_in_aaga(T206, X15, T201, T9) → U30_aaga(T206, X15, T201, T9, qsL_in_a(T206))
qsL_in_a([]) → qsL_out_a([])
U30_aaga(T206, X15, T201, T9, qsL_out_a(T206)) → U31_aaga(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
pS_in_agga(T211, T206, T201, T9) → U32_agga(T211, T206, T201, T9, qsL_in_a(T211))
U32_agga(T211, T206, T201, T9, qsL_out_a(T211)) → U33_agga(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U33_agga(T211, T206, T201, T9, appK_out_ggga(T206, T201, T211, T9)) → pS_out_agga(T211, T206, T201, T9)
U31_aaga(T206, X15, T201, T9, pS_out_agga(X15, T206, T201, T9)) → pD_out_aaga(T206, X15, T201, T9)
U3_ga(T201, T9, pD_out_aaga(X14, X15, T201, T9)) → qsA_out_ga(.(T201, []), T9)
U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_out_gagga(T46, X15, T93, T22, T9)) → pN_out_ggagaga(T23, T45, T93, T46, X15, T22, T9)
U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_out_ggagaga(T23, T45, X14, T46, X15, T22, T9)) → pM_out_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9)
U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_out_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9)) → pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)) → qsA_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2)  =  qsA_in_ga(x1)
[]  =  []
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_gggaaaaa(x1, x2, x3)
U10_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_gggaaaaa(x1, x2, x3, x9)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
pM_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_in_ggaagaaa(x1, x2, x5)
U12_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_ggaagaaa(x1, x2, x5, x9)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_ggaagaaa(x1, x2, x3, x4, x5, x9)
pN_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_in_ggagaga(x1, x2, x4, x6)
U14_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_ggaaagaa(x1, x2, x6)
U26_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_ggaaagaa(x1, x2, x6, x9)
U27_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_ggaaagaa(x1, x2, x3, x4, x6, x9)
pR_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_in_gaggaga(x1, x3, x4, x6)
U28_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_gaggaga(x1, x3, x4, x6, x8)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U29_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_gaggaga(x1, x2, x3, x4, x6, x8)
pO_in_gagga(x1, x2, x3, x4, x5)  =  pO_in_gagga(x1, x3, x4)
U18_gagga(x1, x2, x3, x4, x5, x6)  =  U18_gagga(x1, x3, x4, x6)
U19_gagga(x1, x2, x3, x4, x5, x6)  =  U19_gagga(x1, x2, x3, x4, x6)
appK_in_ggga(x1, x2, x3, x4)  =  appK_in_ggga(x1, x2, x3)
appK_out_ggga(x1, x2, x3, x4)  =  appK_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
pO_out_gagga(x1, x2, x3, x4, x5)  =  pO_out_gagga(x1, x2, x3, x4, x5)
pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U30_aaga(x1, x2, x3, x4, x5)  =  U30_aaga(x3, x5)
qsL_in_a(x1)  =  qsL_in_a
qsL_out_a(x1)  =  qsL_out_a(x1)
U31_aaga(x1, x2, x3, x4, x5)  =  U31_aaga(x1, x3, x5)
pS_in_agga(x1, x2, x3, x4)  =  pS_in_agga(x2, x3)
U32_agga(x1, x2, x3, x4, x5)  =  U32_agga(x2, x3, x5)
U33_agga(x1, x2, x3, x4, x5)  =  U33_agga(x1, x2, x3, x5)
pS_out_agga(x1, x2, x3, x4)  =  pS_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x3, x4, x6, x8)
pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
QSA_IN_GA(x1, x2)  =  QSA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
PB_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGGAAAAA(x1, x2, x3)
U10_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_GGGAAAAA(x1, x2, x3, x9)
LESSE_IN_GG(x1, x2)  =  LESSE_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
U11_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_GGGAAAAA(x1, x2, x3, x9)
PM_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PM_IN_GGAAGAAA(x1, x2, x5)
U12_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_GGAAGAAA(x1, x2, x5, x9)
PARTF_IN_GGAA(x1, x2, x3, x4)  =  PARTF_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
PG_IN_GGGAA(x1, x2, x3, x4, x5)  =  PG_IN_GGGAA(x1, x2, x3)
U16_GGGAA(x1, x2, x3, x4, x5, x6)  =  U16_GGGAA(x1, x2, x3, x6)
U17_GGGAA(x1, x2, x3, x4, x5, x6)  =  U17_GGGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U13_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_GGAAGAAA(x1, x2, x3, x4, x5, x9)
PN_IN_GGAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PN_IN_GGAGAGA(x1, x2, x4, x6)
U14_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GGAGAGA(x1, x2, x4, x6, x8)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
PC_IN_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PC_IN_GGAAAGAA(x1, x2, x6)
U26_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_GGAAAGAA(x1, x2, x6, x9)
U27_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_GGAAAGAA(x1, x2, x3, x4, x6, x9)
PR_IN_GAGGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PR_IN_GAGGAGA(x1, x3, x4, x6)
U28_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_GAGGAGA(x1, x3, x4, x6, x8)
QSH_IN_GA(x1, x2)  =  QSH_IN_GA(x1)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x2, x4)
PI_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PI_IN_GGAAAAA(x1, x2)
U20_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_GGAAAAA(x1, x2, x8)
U21_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_GGAAAAA(x1, x2, x3, x4, x8)
PP_IN_GAGAGA(x1, x2, x3, x4, x5, x6)  =  PP_IN_GAGAGA(x1, x3, x5)
U22_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U22_GAGAGA(x1, x3, x5, x7)
U23_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U23_GAGAGA(x1, x2, x3, x5, x7)
PQ_IN_GAGGA(x1, x2, x3, x4, x5)  =  PQ_IN_GAGGA(x1, x3, x4)
U24_GAGGA(x1, x2, x3, x4, x5, x6)  =  U24_GAGGA(x1, x3, x4, x6)
U25_GAGGA(x1, x2, x3, x4, x5, x6)  =  U25_GAGGA(x1, x2, x3, x4, x6)
APPJ_IN_GGGA(x1, x2, x3, x4)  =  APPJ_IN_GGGA(x1, x2, x3)
U8_GGGA(x1, x2, x3, x4, x5, x6)  =  U8_GGGA(x1, x2, x3, x4, x6)
U29_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_GAGGAGA(x1, x2, x3, x4, x6, x8)
PO_IN_GAGGA(x1, x2, x3, x4, x5)  =  PO_IN_GAGGA(x1, x3, x4)
U18_GAGGA(x1, x2, x3, x4, x5, x6)  =  U18_GAGGA(x1, x3, x4, x6)
U19_GAGGA(x1, x2, x3, x4, x5, x6)  =  U19_GAGGA(x1, x2, x3, x4, x6)
APPK_IN_GGGA(x1, x2, x3, x4)  =  APPK_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6)  =  U9_GGGA(x1, x2, x3, x4, x6)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
PD_IN_AAGA(x1, x2, x3, x4)  =  PD_IN_AAGA(x3)
U30_AAGA(x1, x2, x3, x4, x5)  =  U30_AAGA(x3, x5)
QSL_IN_A(x1)  =  QSL_IN_A
U31_AAGA(x1, x2, x3, x4, x5)  =  U31_AAGA(x1, x3, x5)
PS_IN_AGGA(x1, x2, x3, x4)  =  PS_IN_AGGA(x2, x3)
U32_AGGA(x1, x2, x3, x4, x5)  =  U32_AGGA(x2, x3, x5)
U33_AGGA(x1, x2, x3, x4, x5)  =  U33_AGGA(x1, x2, x3, x5)
U15_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GGAGAGA(x1, x2, x3, x4, x6, x8)

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

(4) Obligation:

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

QSA_IN_GA(.(T22, .(T23, T24)), T9) → U1_GA(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
QSA_IN_GA(.(T22, .(T23, T24)), T9) → PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9)
PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9) → U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9) → LESSE_IN_GG(T22, T23)
LESSE_IN_GG(s(T38), s(T39)) → U4_GG(T38, T39, lessE_in_gg(T38, T39))
LESSE_IN_GG(s(T38), s(T39)) → LESSE_IN_GG(T38, T39)
U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → PM_IN_GGAAGAAA(T22, T24, X49, X50, T23, X14, X15, T9)
PM_IN_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9) → U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
PM_IN_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9) → PARTF_IN_GGAA(T22, T24, T45, T46)
PARTF_IN_GGAA(T62, .(T63, T64), .(T63, X115), X116) → U5_GGAA(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
PARTF_IN_GGAA(T62, .(T63, T64), .(T63, X115), X116) → PG_IN_GGGAA(T62, T63, T64, X115, X116)
PG_IN_GGGAA(T62, T63, T64, X115, X116) → U16_GGGAA(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
PG_IN_GGGAA(T62, T63, T64, X115, X116) → LESSE_IN_GG(T62, T63)
U16_GGGAA(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_GGGAA(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
U16_GGGAA(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → PARTF_IN_GGAA(T62, T64, X115, X116)
PARTF_IN_GGAA(T84, .(T85, T86), X161, .(T85, X162)) → U6_GGAA(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
PARTF_IN_GGAA(T84, .(T85, T86), X161, .(T85, X162)) → PARTF_IN_GGAA(T84, T86, X161, X162)
U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → PN_IN_GGAGAGA(T23, T45, X14, T46, X15, T22, T9)
PN_IN_GGAGAGA(T23, T45, T93, T46, X15, T22, T9) → U14_GGAGAGA(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
PN_IN_GGAGAGA(T23, T45, T93, T46, X15, T22, T9) → QSA_IN_GA(.(T23, T45), T93)
QSA_IN_GA(.(T187, .(T188, T189)), T9) → U2_GA(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
QSA_IN_GA(.(T187, .(T188, T189)), T9) → PC_IN_GGAAAGAA(T187, T189, X285, X286, X14, T188, X15, T9)
PC_IN_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9) → U26_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
PC_IN_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9) → PARTF_IN_GGAA(T187, T189, T193, T194)
U26_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
U26_GGAAAGAA(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → PR_IN_GAGGAGA(T193, X14, T188, T194, X15, T187, T9)
PR_IN_GAGGAGA(T193, T198, T188, T194, X15, T187, T9) → U28_GAGGAGA(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
PR_IN_GAGGAGA(T193, T198, T188, T194, X15, T187, T9) → QSH_IN_GA(T193, T198)
QSH_IN_GA(.(T99, T100), X192) → U7_GA(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
QSH_IN_GA(.(T99, T100), X192) → PI_IN_GGAAAAA(T99, T100, X188, X189, X190, X191, X192)
PI_IN_GGAAAAA(T99, T100, T104, T105, X190, X191, X192) → U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
PI_IN_GGAAAAA(T99, T100, T104, T105, X190, X191, X192) → PARTF_IN_GGAA(T99, T100, T104, T105)
U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → PP_IN_GAGAGA(T104, X190, T105, X191, T99, X192)
PP_IN_GAGAGA(T104, T109, T105, X191, T99, X192) → U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
PP_IN_GAGAGA(T104, T109, T105, X191, T99, X192) → QSH_IN_GA(T104, T109)
U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_GAGAGA(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → PQ_IN_GAGGA(T105, X191, T109, T99, X192)
PQ_IN_GAGGA(T105, T110, T109, T99, X192) → U24_GAGGA(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
PQ_IN_GAGGA(T105, T110, T109, T99, X192) → QSH_IN_GA(T105, T110)
U24_GAGGA(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_GAGGA(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
U24_GAGGA(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → APPJ_IN_GGGA(T109, T99, T110, X192)
APPJ_IN_GGGA(.(T133, T134), T135, T136, .(T133, X228)) → U8_GGGA(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
APPJ_IN_GGGA(.(T133, T134), T135, T136, .(T133, X228)) → APPJ_IN_GGGA(T134, T135, T136, X228)
U28_GAGGAGA(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_GAGGAGA(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
U28_GAGGAGA(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → PO_IN_GAGGA(.(T188, T194), X15, T198, T187, T9)
PO_IN_GAGGA(T46, T94, T93, T22, T9) → U18_GAGGA(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
PO_IN_GAGGA(T46, T94, T93, T22, T9) → QSH_IN_GA(T46, T94)
U18_GAGGA(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_GAGGA(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
U18_GAGGA(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → APPK_IN_GGGA(T93, T22, T94, T9)
APPK_IN_GGGA(.(T165, T166), T167, T168, .(T165, T170)) → U9_GGGA(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
APPK_IN_GGGA(.(T165, T166), T167, T168, .(T165, T170)) → APPK_IN_GGGA(T166, T167, T168, T170)
QSA_IN_GA(.(T201, []), T9) → U3_GA(T201, T9, pD_in_aaga(X14, X15, T201, T9))
QSA_IN_GA(.(T201, []), T9) → PD_IN_AAGA(X14, X15, T201, T9)
PD_IN_AAGA(T206, X15, T201, T9) → U30_AAGA(T206, X15, T201, T9, qsL_in_a(T206))
PD_IN_AAGA(T206, X15, T201, T9) → QSL_IN_A(T206)
U30_AAGA(T206, X15, T201, T9, qsL_out_a(T206)) → U31_AAGA(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
U30_AAGA(T206, X15, T201, T9, qsL_out_a(T206)) → PS_IN_AGGA(X15, T206, T201, T9)
PS_IN_AGGA(T211, T206, T201, T9) → U32_AGGA(T211, T206, T201, T9, qsL_in_a(T211))
PS_IN_AGGA(T211, T206, T201, T9) → QSL_IN_A(T211)
U32_AGGA(T211, T206, T201, T9, qsL_out_a(T211)) → U33_AGGA(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U32_AGGA(T211, T206, T201, T9, qsL_out_a(T211)) → APPK_IN_GGGA(T206, T201, T211, T9)
U14_GGAGAGA(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_GGAGAGA(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U14_GGAGAGA(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → PO_IN_GAGGA(T46, X15, T93, T22, T9)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9) → U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
pM_in_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9) → U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
pN_in_ggagaga(T23, T45, T93, T46, X15, T22, T9) → U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
qsA_in_ga(.(T187, .(T188, T189)), T9) → U2_ga(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
pC_in_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9) → U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
pR_in_gaggaga(T193, T198, T188, T194, X15, T187, T9) → U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
pO_in_gagga(T46, T94, T93, T22, T9) → U18_gagga(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
U18_gagga(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_gagga(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
appK_in_ggga([], T153, T154, .(T153, T154)) → appK_out_ggga([], T153, T154, .(T153, T154))
appK_in_ggga(.(T165, T166), T167, T168, .(T165, T170)) → U9_ggga(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
U9_ggga(T165, T166, T167, T168, T170, appK_out_ggga(T166, T167, T168, T170)) → appK_out_ggga(.(T165, T166), T167, T168, .(T165, T170))
U19_gagga(T46, T94, T93, T22, T9, appK_out_ggga(T93, T22, T94, T9)) → pO_out_gagga(T46, T94, T93, T22, T9)
U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_out_gagga(.(T188, T194), X15, T198, T187, T9)) → pR_out_gaggaga(T193, T198, T188, T194, X15, T187, T9)
U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_out_gaggaga(T193, X14, T188, T194, X15, T187, T9)) → pC_out_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9)
U2_ga(T187, T188, T189, T9, pC_out_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9)) → qsA_out_ga(.(T187, .(T188, T189)), T9)
qsA_in_ga(.(T201, []), T9) → U3_ga(T201, T9, pD_in_aaga(X14, X15, T201, T9))
pD_in_aaga(T206, X15, T201, T9) → U30_aaga(T206, X15, T201, T9, qsL_in_a(T206))
qsL_in_a([]) → qsL_out_a([])
U30_aaga(T206, X15, T201, T9, qsL_out_a(T206)) → U31_aaga(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
pS_in_agga(T211, T206, T201, T9) → U32_agga(T211, T206, T201, T9, qsL_in_a(T211))
U32_agga(T211, T206, T201, T9, qsL_out_a(T211)) → U33_agga(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U33_agga(T211, T206, T201, T9, appK_out_ggga(T206, T201, T211, T9)) → pS_out_agga(T211, T206, T201, T9)
U31_aaga(T206, X15, T201, T9, pS_out_agga(X15, T206, T201, T9)) → pD_out_aaga(T206, X15, T201, T9)
U3_ga(T201, T9, pD_out_aaga(X14, X15, T201, T9)) → qsA_out_ga(.(T201, []), T9)
U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_out_gagga(T46, X15, T93, T22, T9)) → pN_out_ggagaga(T23, T45, T93, T46, X15, T22, T9)
U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_out_ggagaga(T23, T45, X14, T46, X15, T22, T9)) → pM_out_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9)
U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_out_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9)) → pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)) → qsA_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2)  =  qsA_in_ga(x1)
[]  =  []
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_gggaaaaa(x1, x2, x3)
U10_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_gggaaaaa(x1, x2, x3, x9)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
pM_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_in_ggaagaaa(x1, x2, x5)
U12_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_ggaagaaa(x1, x2, x5, x9)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_ggaagaaa(x1, x2, x3, x4, x5, x9)
pN_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_in_ggagaga(x1, x2, x4, x6)
U14_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_ggaaagaa(x1, x2, x6)
U26_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_ggaaagaa(x1, x2, x6, x9)
U27_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_ggaaagaa(x1, x2, x3, x4, x6, x9)
pR_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_in_gaggaga(x1, x3, x4, x6)
U28_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_gaggaga(x1, x3, x4, x6, x8)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U29_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_gaggaga(x1, x2, x3, x4, x6, x8)
pO_in_gagga(x1, x2, x3, x4, x5)  =  pO_in_gagga(x1, x3, x4)
U18_gagga(x1, x2, x3, x4, x5, x6)  =  U18_gagga(x1, x3, x4, x6)
U19_gagga(x1, x2, x3, x4, x5, x6)  =  U19_gagga(x1, x2, x3, x4, x6)
appK_in_ggga(x1, x2, x3, x4)  =  appK_in_ggga(x1, x2, x3)
appK_out_ggga(x1, x2, x3, x4)  =  appK_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
pO_out_gagga(x1, x2, x3, x4, x5)  =  pO_out_gagga(x1, x2, x3, x4, x5)
pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U30_aaga(x1, x2, x3, x4, x5)  =  U30_aaga(x3, x5)
qsL_in_a(x1)  =  qsL_in_a
qsL_out_a(x1)  =  qsL_out_a(x1)
U31_aaga(x1, x2, x3, x4, x5)  =  U31_aaga(x1, x3, x5)
pS_in_agga(x1, x2, x3, x4)  =  pS_in_agga(x2, x3)
U32_agga(x1, x2, x3, x4, x5)  =  U32_agga(x2, x3, x5)
U33_agga(x1, x2, x3, x4, x5)  =  U33_agga(x1, x2, x3, x5)
pS_out_agga(x1, x2, x3, x4)  =  pS_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x3, x4, x6, x8)
pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
QSA_IN_GA(x1, x2)  =  QSA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
PB_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGGAAAAA(x1, x2, x3)
U10_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_GGGAAAAA(x1, x2, x3, x9)
LESSE_IN_GG(x1, x2)  =  LESSE_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
U11_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_GGGAAAAA(x1, x2, x3, x9)
PM_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PM_IN_GGAAGAAA(x1, x2, x5)
U12_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_GGAAGAAA(x1, x2, x5, x9)
PARTF_IN_GGAA(x1, x2, x3, x4)  =  PARTF_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
PG_IN_GGGAA(x1, x2, x3, x4, x5)  =  PG_IN_GGGAA(x1, x2, x3)
U16_GGGAA(x1, x2, x3, x4, x5, x6)  =  U16_GGGAA(x1, x2, x3, x6)
U17_GGGAA(x1, x2, x3, x4, x5, x6)  =  U17_GGGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U13_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_GGAAGAAA(x1, x2, x3, x4, x5, x9)
PN_IN_GGAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PN_IN_GGAGAGA(x1, x2, x4, x6)
U14_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GGAGAGA(x1, x2, x4, x6, x8)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
PC_IN_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PC_IN_GGAAAGAA(x1, x2, x6)
U26_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_GGAAAGAA(x1, x2, x6, x9)
U27_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_GGAAAGAA(x1, x2, x3, x4, x6, x9)
PR_IN_GAGGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PR_IN_GAGGAGA(x1, x3, x4, x6)
U28_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_GAGGAGA(x1, x3, x4, x6, x8)
QSH_IN_GA(x1, x2)  =  QSH_IN_GA(x1)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x2, x4)
PI_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PI_IN_GGAAAAA(x1, x2)
U20_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_GGAAAAA(x1, x2, x8)
U21_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_GGAAAAA(x1, x2, x3, x4, x8)
PP_IN_GAGAGA(x1, x2, x3, x4, x5, x6)  =  PP_IN_GAGAGA(x1, x3, x5)
U22_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U22_GAGAGA(x1, x3, x5, x7)
U23_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U23_GAGAGA(x1, x2, x3, x5, x7)
PQ_IN_GAGGA(x1, x2, x3, x4, x5)  =  PQ_IN_GAGGA(x1, x3, x4)
U24_GAGGA(x1, x2, x3, x4, x5, x6)  =  U24_GAGGA(x1, x3, x4, x6)
U25_GAGGA(x1, x2, x3, x4, x5, x6)  =  U25_GAGGA(x1, x2, x3, x4, x6)
APPJ_IN_GGGA(x1, x2, x3, x4)  =  APPJ_IN_GGGA(x1, x2, x3)
U8_GGGA(x1, x2, x3, x4, x5, x6)  =  U8_GGGA(x1, x2, x3, x4, x6)
U29_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_GAGGAGA(x1, x2, x3, x4, x6, x8)
PO_IN_GAGGA(x1, x2, x3, x4, x5)  =  PO_IN_GAGGA(x1, x3, x4)
U18_GAGGA(x1, x2, x3, x4, x5, x6)  =  U18_GAGGA(x1, x3, x4, x6)
U19_GAGGA(x1, x2, x3, x4, x5, x6)  =  U19_GAGGA(x1, x2, x3, x4, x6)
APPK_IN_GGGA(x1, x2, x3, x4)  =  APPK_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6)  =  U9_GGGA(x1, x2, x3, x4, x6)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
PD_IN_AAGA(x1, x2, x3, x4)  =  PD_IN_AAGA(x3)
U30_AAGA(x1, x2, x3, x4, x5)  =  U30_AAGA(x3, x5)
QSL_IN_A(x1)  =  QSL_IN_A
U31_AAGA(x1, x2, x3, x4, x5)  =  U31_AAGA(x1, x3, x5)
PS_IN_AGGA(x1, x2, x3, x4)  =  PS_IN_AGGA(x2, x3)
U32_AGGA(x1, x2, x3, x4, x5)  =  U32_AGGA(x2, x3, x5)
U33_AGGA(x1, x2, x3, x4, x5)  =  U33_AGGA(x1, x2, x3, x5)
U15_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GGAGAGA(x1, x2, x3, x4, x6, x8)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 46 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPK_IN_GGGA(.(T165, T166), T167, T168, .(T165, T170)) → APPK_IN_GGGA(T166, T167, T168, T170)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9) → U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
pM_in_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9) → U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
pN_in_ggagaga(T23, T45, T93, T46, X15, T22, T9) → U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
qsA_in_ga(.(T187, .(T188, T189)), T9) → U2_ga(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
pC_in_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9) → U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
pR_in_gaggaga(T193, T198, T188, T194, X15, T187, T9) → U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
pO_in_gagga(T46, T94, T93, T22, T9) → U18_gagga(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
U18_gagga(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_gagga(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
appK_in_ggga([], T153, T154, .(T153, T154)) → appK_out_ggga([], T153, T154, .(T153, T154))
appK_in_ggga(.(T165, T166), T167, T168, .(T165, T170)) → U9_ggga(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
U9_ggga(T165, T166, T167, T168, T170, appK_out_ggga(T166, T167, T168, T170)) → appK_out_ggga(.(T165, T166), T167, T168, .(T165, T170))
U19_gagga(T46, T94, T93, T22, T9, appK_out_ggga(T93, T22, T94, T9)) → pO_out_gagga(T46, T94, T93, T22, T9)
U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_out_gagga(.(T188, T194), X15, T198, T187, T9)) → pR_out_gaggaga(T193, T198, T188, T194, X15, T187, T9)
U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_out_gaggaga(T193, X14, T188, T194, X15, T187, T9)) → pC_out_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9)
U2_ga(T187, T188, T189, T9, pC_out_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9)) → qsA_out_ga(.(T187, .(T188, T189)), T9)
qsA_in_ga(.(T201, []), T9) → U3_ga(T201, T9, pD_in_aaga(X14, X15, T201, T9))
pD_in_aaga(T206, X15, T201, T9) → U30_aaga(T206, X15, T201, T9, qsL_in_a(T206))
qsL_in_a([]) → qsL_out_a([])
U30_aaga(T206, X15, T201, T9, qsL_out_a(T206)) → U31_aaga(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
pS_in_agga(T211, T206, T201, T9) → U32_agga(T211, T206, T201, T9, qsL_in_a(T211))
U32_agga(T211, T206, T201, T9, qsL_out_a(T211)) → U33_agga(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U33_agga(T211, T206, T201, T9, appK_out_ggga(T206, T201, T211, T9)) → pS_out_agga(T211, T206, T201, T9)
U31_aaga(T206, X15, T201, T9, pS_out_agga(X15, T206, T201, T9)) → pD_out_aaga(T206, X15, T201, T9)
U3_ga(T201, T9, pD_out_aaga(X14, X15, T201, T9)) → qsA_out_ga(.(T201, []), T9)
U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_out_gagga(T46, X15, T93, T22, T9)) → pN_out_ggagaga(T23, T45, T93, T46, X15, T22, T9)
U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_out_ggagaga(T23, T45, X14, T46, X15, T22, T9)) → pM_out_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9)
U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_out_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9)) → pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)) → qsA_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2)  =  qsA_in_ga(x1)
[]  =  []
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_gggaaaaa(x1, x2, x3)
U10_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_gggaaaaa(x1, x2, x3, x9)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
pM_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_in_ggaagaaa(x1, x2, x5)
U12_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_ggaagaaa(x1, x2, x5, x9)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_ggaagaaa(x1, x2, x3, x4, x5, x9)
pN_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_in_ggagaga(x1, x2, x4, x6)
U14_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_ggaaagaa(x1, x2, x6)
U26_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_ggaaagaa(x1, x2, x6, x9)
U27_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_ggaaagaa(x1, x2, x3, x4, x6, x9)
pR_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_in_gaggaga(x1, x3, x4, x6)
U28_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_gaggaga(x1, x3, x4, x6, x8)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U29_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_gaggaga(x1, x2, x3, x4, x6, x8)
pO_in_gagga(x1, x2, x3, x4, x5)  =  pO_in_gagga(x1, x3, x4)
U18_gagga(x1, x2, x3, x4, x5, x6)  =  U18_gagga(x1, x3, x4, x6)
U19_gagga(x1, x2, x3, x4, x5, x6)  =  U19_gagga(x1, x2, x3, x4, x6)
appK_in_ggga(x1, x2, x3, x4)  =  appK_in_ggga(x1, x2, x3)
appK_out_ggga(x1, x2, x3, x4)  =  appK_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
pO_out_gagga(x1, x2, x3, x4, x5)  =  pO_out_gagga(x1, x2, x3, x4, x5)
pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U30_aaga(x1, x2, x3, x4, x5)  =  U30_aaga(x3, x5)
qsL_in_a(x1)  =  qsL_in_a
qsL_out_a(x1)  =  qsL_out_a(x1)
U31_aaga(x1, x2, x3, x4, x5)  =  U31_aaga(x1, x3, x5)
pS_in_agga(x1, x2, x3, x4)  =  pS_in_agga(x2, x3)
U32_agga(x1, x2, x3, x4, x5)  =  U32_agga(x2, x3, x5)
U33_agga(x1, x2, x3, x4, x5)  =  U33_agga(x1, x2, x3, x5)
pS_out_agga(x1, x2, x3, x4)  =  pS_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x3, x4, x6, x8)
pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
APPK_IN_GGGA(x1, x2, x3, x4)  =  APPK_IN_GGGA(x1, x2, x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APPK_IN_GGGA(.(T165, T166), T167, T168, .(T165, T170)) → APPK_IN_GGGA(T166, T167, T168, T170)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APPK_IN_GGGA(.(T165, T166), T167, T168) → APPK_IN_GGGA(T166, T167, T168)

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:

• APPK_IN_GGGA(.(T165, T166), T167, T168) → APPK_IN_GGGA(T166, T167, T168)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(13) YES

(14) Obligation:

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

APPJ_IN_GGGA(.(T133, T134), T135, T136, .(T133, X228)) → APPJ_IN_GGGA(T134, T135, T136, X228)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9) → U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
pM_in_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9) → U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
pN_in_ggagaga(T23, T45, T93, T46, X15, T22, T9) → U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
qsA_in_ga(.(T187, .(T188, T189)), T9) → U2_ga(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
pC_in_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9) → U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
pR_in_gaggaga(T193, T198, T188, T194, X15, T187, T9) → U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
pO_in_gagga(T46, T94, T93, T22, T9) → U18_gagga(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
U18_gagga(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_gagga(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
appK_in_ggga([], T153, T154, .(T153, T154)) → appK_out_ggga([], T153, T154, .(T153, T154))
appK_in_ggga(.(T165, T166), T167, T168, .(T165, T170)) → U9_ggga(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
U9_ggga(T165, T166, T167, T168, T170, appK_out_ggga(T166, T167, T168, T170)) → appK_out_ggga(.(T165, T166), T167, T168, .(T165, T170))
U19_gagga(T46, T94, T93, T22, T9, appK_out_ggga(T93, T22, T94, T9)) → pO_out_gagga(T46, T94, T93, T22, T9)
U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_out_gagga(.(T188, T194), X15, T198, T187, T9)) → pR_out_gaggaga(T193, T198, T188, T194, X15, T187, T9)
U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_out_gaggaga(T193, X14, T188, T194, X15, T187, T9)) → pC_out_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9)
U2_ga(T187, T188, T189, T9, pC_out_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9)) → qsA_out_ga(.(T187, .(T188, T189)), T9)
qsA_in_ga(.(T201, []), T9) → U3_ga(T201, T9, pD_in_aaga(X14, X15, T201, T9))
pD_in_aaga(T206, X15, T201, T9) → U30_aaga(T206, X15, T201, T9, qsL_in_a(T206))
qsL_in_a([]) → qsL_out_a([])
U30_aaga(T206, X15, T201, T9, qsL_out_a(T206)) → U31_aaga(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
pS_in_agga(T211, T206, T201, T9) → U32_agga(T211, T206, T201, T9, qsL_in_a(T211))
U32_agga(T211, T206, T201, T9, qsL_out_a(T211)) → U33_agga(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U33_agga(T211, T206, T201, T9, appK_out_ggga(T206, T201, T211, T9)) → pS_out_agga(T211, T206, T201, T9)
U31_aaga(T206, X15, T201, T9, pS_out_agga(X15, T206, T201, T9)) → pD_out_aaga(T206, X15, T201, T9)
U3_ga(T201, T9, pD_out_aaga(X14, X15, T201, T9)) → qsA_out_ga(.(T201, []), T9)
U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_out_gagga(T46, X15, T93, T22, T9)) → pN_out_ggagaga(T23, T45, T93, T46, X15, T22, T9)
U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_out_ggagaga(T23, T45, X14, T46, X15, T22, T9)) → pM_out_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9)
U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_out_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9)) → pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)) → qsA_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2)  =  qsA_in_ga(x1)
[]  =  []
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_gggaaaaa(x1, x2, x3)
U10_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_gggaaaaa(x1, x2, x3, x9)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
pM_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_in_ggaagaaa(x1, x2, x5)
U12_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_ggaagaaa(x1, x2, x5, x9)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_ggaagaaa(x1, x2, x3, x4, x5, x9)
pN_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_in_ggagaga(x1, x2, x4, x6)
U14_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_ggaaagaa(x1, x2, x6)
U26_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_ggaaagaa(x1, x2, x6, x9)
U27_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_ggaaagaa(x1, x2, x3, x4, x6, x9)
pR_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_in_gaggaga(x1, x3, x4, x6)
U28_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_gaggaga(x1, x3, x4, x6, x8)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U29_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_gaggaga(x1, x2, x3, x4, x6, x8)
pO_in_gagga(x1, x2, x3, x4, x5)  =  pO_in_gagga(x1, x3, x4)
U18_gagga(x1, x2, x3, x4, x5, x6)  =  U18_gagga(x1, x3, x4, x6)
U19_gagga(x1, x2, x3, x4, x5, x6)  =  U19_gagga(x1, x2, x3, x4, x6)
appK_in_ggga(x1, x2, x3, x4)  =  appK_in_ggga(x1, x2, x3)
appK_out_ggga(x1, x2, x3, x4)  =  appK_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
pO_out_gagga(x1, x2, x3, x4, x5)  =  pO_out_gagga(x1, x2, x3, x4, x5)
pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U30_aaga(x1, x2, x3, x4, x5)  =  U30_aaga(x3, x5)
qsL_in_a(x1)  =  qsL_in_a
qsL_out_a(x1)  =  qsL_out_a(x1)
U31_aaga(x1, x2, x3, x4, x5)  =  U31_aaga(x1, x3, x5)
pS_in_agga(x1, x2, x3, x4)  =  pS_in_agga(x2, x3)
U32_agga(x1, x2, x3, x4, x5)  =  U32_agga(x2, x3, x5)
U33_agga(x1, x2, x3, x4, x5)  =  U33_agga(x1, x2, x3, x5)
pS_out_agga(x1, x2, x3, x4)  =  pS_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x3, x4, x6, x8)
pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
APPJ_IN_GGGA(x1, x2, x3, x4)  =  APPJ_IN_GGGA(x1, x2, x3)

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:

APPJ_IN_GGGA(.(T133, T134), T135, T136, .(T133, X228)) → APPJ_IN_GGGA(T134, T135, T136, X228)

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APPJ_IN_GGGA(.(T133, T134), T135, T136) → APPJ_IN_GGGA(T134, T135, T136)

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:

• APPJ_IN_GGGA(.(T133, T134), T135, T136) → APPJ_IN_GGGA(T134, T135, T136)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(20) YES

(21) Obligation:

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

LESSE_IN_GG(s(T38), s(T39)) → LESSE_IN_GG(T38, T39)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9) → U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
pM_in_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9) → U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
pN_in_ggagaga(T23, T45, T93, T46, X15, T22, T9) → U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
qsA_in_ga(.(T187, .(T188, T189)), T9) → U2_ga(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
pC_in_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9) → U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
pR_in_gaggaga(T193, T198, T188, T194, X15, T187, T9) → U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
pO_in_gagga(T46, T94, T93, T22, T9) → U18_gagga(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
U18_gagga(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_gagga(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
appK_in_ggga([], T153, T154, .(T153, T154)) → appK_out_ggga([], T153, T154, .(T153, T154))
appK_in_ggga(.(T165, T166), T167, T168, .(T165, T170)) → U9_ggga(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
U9_ggga(T165, T166, T167, T168, T170, appK_out_ggga(T166, T167, T168, T170)) → appK_out_ggga(.(T165, T166), T167, T168, .(T165, T170))
U19_gagga(T46, T94, T93, T22, T9, appK_out_ggga(T93, T22, T94, T9)) → pO_out_gagga(T46, T94, T93, T22, T9)
U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_out_gagga(.(T188, T194), X15, T198, T187, T9)) → pR_out_gaggaga(T193, T198, T188, T194, X15, T187, T9)
U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_out_gaggaga(T193, X14, T188, T194, X15, T187, T9)) → pC_out_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9)
U2_ga(T187, T188, T189, T9, pC_out_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9)) → qsA_out_ga(.(T187, .(T188, T189)), T9)
qsA_in_ga(.(T201, []), T9) → U3_ga(T201, T9, pD_in_aaga(X14, X15, T201, T9))
pD_in_aaga(T206, X15, T201, T9) → U30_aaga(T206, X15, T201, T9, qsL_in_a(T206))
qsL_in_a([]) → qsL_out_a([])
U30_aaga(T206, X15, T201, T9, qsL_out_a(T206)) → U31_aaga(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
pS_in_agga(T211, T206, T201, T9) → U32_agga(T211, T206, T201, T9, qsL_in_a(T211))
U32_agga(T211, T206, T201, T9, qsL_out_a(T211)) → U33_agga(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U33_agga(T211, T206, T201, T9, appK_out_ggga(T206, T201, T211, T9)) → pS_out_agga(T211, T206, T201, T9)
U31_aaga(T206, X15, T201, T9, pS_out_agga(X15, T206, T201, T9)) → pD_out_aaga(T206, X15, T201, T9)
U3_ga(T201, T9, pD_out_aaga(X14, X15, T201, T9)) → qsA_out_ga(.(T201, []), T9)
U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_out_gagga(T46, X15, T93, T22, T9)) → pN_out_ggagaga(T23, T45, T93, T46, X15, T22, T9)
U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_out_ggagaga(T23, T45, X14, T46, X15, T22, T9)) → pM_out_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9)
U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_out_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9)) → pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)) → qsA_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2)  =  qsA_in_ga(x1)
[]  =  []
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_gggaaaaa(x1, x2, x3)
U10_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_gggaaaaa(x1, x2, x3, x9)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
pM_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_in_ggaagaaa(x1, x2, x5)
U12_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_ggaagaaa(x1, x2, x5, x9)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_ggaagaaa(x1, x2, x3, x4, x5, x9)
pN_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_in_ggagaga(x1, x2, x4, x6)
U14_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_ggaaagaa(x1, x2, x6)
U26_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_ggaaagaa(x1, x2, x6, x9)
U27_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_ggaaagaa(x1, x2, x3, x4, x6, x9)
pR_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_in_gaggaga(x1, x3, x4, x6)
U28_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_gaggaga(x1, x3, x4, x6, x8)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U29_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_gaggaga(x1, x2, x3, x4, x6, x8)
pO_in_gagga(x1, x2, x3, x4, x5)  =  pO_in_gagga(x1, x3, x4)
U18_gagga(x1, x2, x3, x4, x5, x6)  =  U18_gagga(x1, x3, x4, x6)
U19_gagga(x1, x2, x3, x4, x5, x6)  =  U19_gagga(x1, x2, x3, x4, x6)
appK_in_ggga(x1, x2, x3, x4)  =  appK_in_ggga(x1, x2, x3)
appK_out_ggga(x1, x2, x3, x4)  =  appK_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
pO_out_gagga(x1, x2, x3, x4, x5)  =  pO_out_gagga(x1, x2, x3, x4, x5)
pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U30_aaga(x1, x2, x3, x4, x5)  =  U30_aaga(x3, x5)
qsL_in_a(x1)  =  qsL_in_a
qsL_out_a(x1)  =  qsL_out_a(x1)
U31_aaga(x1, x2, x3, x4, x5)  =  U31_aaga(x1, x3, x5)
pS_in_agga(x1, x2, x3, x4)  =  pS_in_agga(x2, x3)
U32_agga(x1, x2, x3, x4, x5)  =  U32_agga(x2, x3, x5)
U33_agga(x1, x2, x3, x4, x5)  =  U33_agga(x1, x2, x3, x5)
pS_out_agga(x1, x2, x3, x4)  =  pS_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x3, x4, x6, x8)
pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
LESSE_IN_GG(x1, x2)  =  LESSE_IN_GG(x1, x2)

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:

LESSE_IN_GG(s(T38), s(T39)) → LESSE_IN_GG(T38, T39)

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

(24) PiDPToQDPProof (EQUIVALENT 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:

LESSE_IN_GG(s(T38), s(T39)) → LESSE_IN_GG(T38, T39)

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

(26) 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:

• LESSE_IN_GG(s(T38), s(T39)) → LESSE_IN_GG(T38, T39)
  The graph contains the following edges 1 > 1, 2 > 2

(27) YES

(28) Obligation:

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

PG_IN_GGGAA(T62, T63, T64, X115, X116) → U16_GGGAA(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_GGGAA(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → PARTF_IN_GGAA(T62, T64, X115, X116)
PARTF_IN_GGAA(T62, .(T63, T64), .(T63, X115), X116) → PG_IN_GGGAA(T62, T63, T64, X115, X116)
PARTF_IN_GGAA(T84, .(T85, T86), X161, .(T85, X162)) → PARTF_IN_GGAA(T84, T86, X161, X162)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9) → U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
pM_in_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9) → U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
pN_in_ggagaga(T23, T45, T93, T46, X15, T22, T9) → U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
qsA_in_ga(.(T187, .(T188, T189)), T9) → U2_ga(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
pC_in_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9) → U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
pR_in_gaggaga(T193, T198, T188, T194, X15, T187, T9) → U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
pO_in_gagga(T46, T94, T93, T22, T9) → U18_gagga(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
U18_gagga(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_gagga(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
appK_in_ggga([], T153, T154, .(T153, T154)) → appK_out_ggga([], T153, T154, .(T153, T154))
appK_in_ggga(.(T165, T166), T167, T168, .(T165, T170)) → U9_ggga(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
U9_ggga(T165, T166, T167, T168, T170, appK_out_ggga(T166, T167, T168, T170)) → appK_out_ggga(.(T165, T166), T167, T168, .(T165, T170))
U19_gagga(T46, T94, T93, T22, T9, appK_out_ggga(T93, T22, T94, T9)) → pO_out_gagga(T46, T94, T93, T22, T9)
U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_out_gagga(.(T188, T194), X15, T198, T187, T9)) → pR_out_gaggaga(T193, T198, T188, T194, X15, T187, T9)
U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_out_gaggaga(T193, X14, T188, T194, X15, T187, T9)) → pC_out_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9)
U2_ga(T187, T188, T189, T9, pC_out_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9)) → qsA_out_ga(.(T187, .(T188, T189)), T9)
qsA_in_ga(.(T201, []), T9) → U3_ga(T201, T9, pD_in_aaga(X14, X15, T201, T9))
pD_in_aaga(T206, X15, T201, T9) → U30_aaga(T206, X15, T201, T9, qsL_in_a(T206))
qsL_in_a([]) → qsL_out_a([])
U30_aaga(T206, X15, T201, T9, qsL_out_a(T206)) → U31_aaga(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
pS_in_agga(T211, T206, T201, T9) → U32_agga(T211, T206, T201, T9, qsL_in_a(T211))
U32_agga(T211, T206, T201, T9, qsL_out_a(T211)) → U33_agga(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U33_agga(T211, T206, T201, T9, appK_out_ggga(T206, T201, T211, T9)) → pS_out_agga(T211, T206, T201, T9)
U31_aaga(T206, X15, T201, T9, pS_out_agga(X15, T206, T201, T9)) → pD_out_aaga(T206, X15, T201, T9)
U3_ga(T201, T9, pD_out_aaga(X14, X15, T201, T9)) → qsA_out_ga(.(T201, []), T9)
U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_out_gagga(T46, X15, T93, T22, T9)) → pN_out_ggagaga(T23, T45, T93, T46, X15, T22, T9)
U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_out_ggagaga(T23, T45, X14, T46, X15, T22, T9)) → pM_out_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9)
U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_out_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9)) → pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)) → qsA_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2)  =  qsA_in_ga(x1)
[]  =  []
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_gggaaaaa(x1, x2, x3)
U10_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_gggaaaaa(x1, x2, x3, x9)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
pM_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_in_ggaagaaa(x1, x2, x5)
U12_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_ggaagaaa(x1, x2, x5, x9)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_ggaagaaa(x1, x2, x3, x4, x5, x9)
pN_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_in_ggagaga(x1, x2, x4, x6)
U14_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_ggaaagaa(x1, x2, x6)
U26_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_ggaaagaa(x1, x2, x6, x9)
U27_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_ggaaagaa(x1, x2, x3, x4, x6, x9)
pR_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_in_gaggaga(x1, x3, x4, x6)
U28_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_gaggaga(x1, x3, x4, x6, x8)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U29_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_gaggaga(x1, x2, x3, x4, x6, x8)
pO_in_gagga(x1, x2, x3, x4, x5)  =  pO_in_gagga(x1, x3, x4)
U18_gagga(x1, x2, x3, x4, x5, x6)  =  U18_gagga(x1, x3, x4, x6)
U19_gagga(x1, x2, x3, x4, x5, x6)  =  U19_gagga(x1, x2, x3, x4, x6)
appK_in_ggga(x1, x2, x3, x4)  =  appK_in_ggga(x1, x2, x3)
appK_out_ggga(x1, x2, x3, x4)  =  appK_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
pO_out_gagga(x1, x2, x3, x4, x5)  =  pO_out_gagga(x1, x2, x3, x4, x5)
pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U30_aaga(x1, x2, x3, x4, x5)  =  U30_aaga(x3, x5)
qsL_in_a(x1)  =  qsL_in_a
qsL_out_a(x1)  =  qsL_out_a(x1)
U31_aaga(x1, x2, x3, x4, x5)  =  U31_aaga(x1, x3, x5)
pS_in_agga(x1, x2, x3, x4)  =  pS_in_agga(x2, x3)
U32_agga(x1, x2, x3, x4, x5)  =  U32_agga(x2, x3, x5)
U33_agga(x1, x2, x3, x4, x5)  =  U33_agga(x1, x2, x3, x5)
pS_out_agga(x1, x2, x3, x4)  =  pS_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x3, x4, x6, x8)
pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
PARTF_IN_GGAA(x1, x2, x3, x4)  =  PARTF_IN_GGAA(x1, x2)
PG_IN_GGGAA(x1, x2, x3, x4, x5)  =  PG_IN_GGGAA(x1, x2, x3)
U16_GGGAA(x1, x2, x3, x4, x5, x6)  =  U16_GGGAA(x1, x2, x3, x6)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

PG_IN_GGGAA(T62, T63, T64, X115, X116) → U16_GGGAA(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_GGGAA(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → PARTF_IN_GGAA(T62, T64, X115, X116)
PARTF_IN_GGAA(T62, .(T63, T64), .(T63, X115), X116) → PG_IN_GGGAA(T62, T63, T64, X115, X116)
PARTF_IN_GGAA(T84, .(T85, T86), X161, .(T85, X162)) → PARTF_IN_GGAA(T84, T86, X161, X162)

The TRS R consists of the following rules:

lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
PARTF_IN_GGAA(x1, x2, x3, x4)  =  PARTF_IN_GGAA(x1, x2)
PG_IN_GGGAA(x1, x2, x3, x4, x5)  =  PG_IN_GGGAA(x1, x2, x3)
U16_GGGAA(x1, x2, x3, x4, x5, x6)  =  U16_GGGAA(x1, x2, x3, x6)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

PG_IN_GGGAA(T62, T63, T64) → U16_GGGAA(T62, T63, T64, lessE_in_gg(T62, T63))
U16_GGGAA(T62, T63, T64, lessE_out_gg(T62, T63)) → PARTF_IN_GGAA(T62, T64)
PARTF_IN_GGAA(T62, .(T63, T64)) → PG_IN_GGGAA(T62, T63, T64)
PARTF_IN_GGAA(T84, .(T85, T86)) → PARTF_IN_GGAA(T84, T86)

The TRS R consists of the following rules:

lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))

The set Q consists of the following terms:

lessE_in_gg(x0, x1)
U4_gg(x0, x1, x2)

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

(33) 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:

• U16_GGGAA(T62, T63, T64, lessE_out_gg(T62, T63)) → PARTF_IN_GGAA(T62, T64)
  The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

• PARTF_IN_GGAA(T62, .(T63, T64)) → PG_IN_GGGAA(T62, T63, T64)
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

• PARTF_IN_GGAA(T84, .(T85, T86)) → PARTF_IN_GGAA(T84, T86)
  The graph contains the following edges 1 >= 1, 2 > 2

• PG_IN_GGGAA(T62, T63, T64) → U16_GGGAA(T62, T63, T64, lessE_in_gg(T62, T63))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

(34) YES

(35) Obligation:

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

PI_IN_GGAAAAA(T99, T100, T104, T105, X190, X191, X192) → U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → PP_IN_GAGAGA(T104, X190, T105, X191, T99, X192)
PP_IN_GAGAGA(T104, T109, T105, X191, T99, X192) → U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → PQ_IN_GAGGA(T105, X191, T109, T99, X192)
PQ_IN_GAGGA(T105, T110, T109, T99, X192) → QSH_IN_GA(T105, T110)
QSH_IN_GA(.(T99, T100), X192) → PI_IN_GGAAAAA(T99, T100, X188, X189, X190, X191, X192)
PP_IN_GAGAGA(T104, T109, T105, X191, T99, X192) → QSH_IN_GA(T104, T109)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9) → U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
pM_in_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9) → U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
pN_in_ggagaga(T23, T45, T93, T46, X15, T22, T9) → U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
qsA_in_ga(.(T187, .(T188, T189)), T9) → U2_ga(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
pC_in_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9) → U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
pR_in_gaggaga(T193, T198, T188, T194, X15, T187, T9) → U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
pO_in_gagga(T46, T94, T93, T22, T9) → U18_gagga(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
U18_gagga(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_gagga(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
appK_in_ggga([], T153, T154, .(T153, T154)) → appK_out_ggga([], T153, T154, .(T153, T154))
appK_in_ggga(.(T165, T166), T167, T168, .(T165, T170)) → U9_ggga(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
U9_ggga(T165, T166, T167, T168, T170, appK_out_ggga(T166, T167, T168, T170)) → appK_out_ggga(.(T165, T166), T167, T168, .(T165, T170))
U19_gagga(T46, T94, T93, T22, T9, appK_out_ggga(T93, T22, T94, T9)) → pO_out_gagga(T46, T94, T93, T22, T9)
U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_out_gagga(.(T188, T194), X15, T198, T187, T9)) → pR_out_gaggaga(T193, T198, T188, T194, X15, T187, T9)
U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_out_gaggaga(T193, X14, T188, T194, X15, T187, T9)) → pC_out_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9)
U2_ga(T187, T188, T189, T9, pC_out_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9)) → qsA_out_ga(.(T187, .(T188, T189)), T9)
qsA_in_ga(.(T201, []), T9) → U3_ga(T201, T9, pD_in_aaga(X14, X15, T201, T9))
pD_in_aaga(T206, X15, T201, T9) → U30_aaga(T206, X15, T201, T9, qsL_in_a(T206))
qsL_in_a([]) → qsL_out_a([])
U30_aaga(T206, X15, T201, T9, qsL_out_a(T206)) → U31_aaga(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
pS_in_agga(T211, T206, T201, T9) → U32_agga(T211, T206, T201, T9, qsL_in_a(T211))
U32_agga(T211, T206, T201, T9, qsL_out_a(T211)) → U33_agga(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U33_agga(T211, T206, T201, T9, appK_out_ggga(T206, T201, T211, T9)) → pS_out_agga(T211, T206, T201, T9)
U31_aaga(T206, X15, T201, T9, pS_out_agga(X15, T206, T201, T9)) → pD_out_aaga(T206, X15, T201, T9)
U3_ga(T201, T9, pD_out_aaga(X14, X15, T201, T9)) → qsA_out_ga(.(T201, []), T9)
U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_out_gagga(T46, X15, T93, T22, T9)) → pN_out_ggagaga(T23, T45, T93, T46, X15, T22, T9)
U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_out_ggagaga(T23, T45, X14, T46, X15, T22, T9)) → pM_out_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9)
U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_out_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9)) → pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)) → qsA_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2)  =  qsA_in_ga(x1)
[]  =  []
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_gggaaaaa(x1, x2, x3)
U10_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_gggaaaaa(x1, x2, x3, x9)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
pM_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_in_ggaagaaa(x1, x2, x5)
U12_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_ggaagaaa(x1, x2, x5, x9)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_ggaagaaa(x1, x2, x3, x4, x5, x9)
pN_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_in_ggagaga(x1, x2, x4, x6)
U14_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_ggaaagaa(x1, x2, x6)
U26_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_ggaaagaa(x1, x2, x6, x9)
U27_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_ggaaagaa(x1, x2, x3, x4, x6, x9)
pR_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_in_gaggaga(x1, x3, x4, x6)
U28_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_gaggaga(x1, x3, x4, x6, x8)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U29_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_gaggaga(x1, x2, x3, x4, x6, x8)
pO_in_gagga(x1, x2, x3, x4, x5)  =  pO_in_gagga(x1, x3, x4)
U18_gagga(x1, x2, x3, x4, x5, x6)  =  U18_gagga(x1, x3, x4, x6)
U19_gagga(x1, x2, x3, x4, x5, x6)  =  U19_gagga(x1, x2, x3, x4, x6)
appK_in_ggga(x1, x2, x3, x4)  =  appK_in_ggga(x1, x2, x3)
appK_out_ggga(x1, x2, x3, x4)  =  appK_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
pO_out_gagga(x1, x2, x3, x4, x5)  =  pO_out_gagga(x1, x2, x3, x4, x5)
pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U30_aaga(x1, x2, x3, x4, x5)  =  U30_aaga(x3, x5)
qsL_in_a(x1)  =  qsL_in_a
qsL_out_a(x1)  =  qsL_out_a(x1)
U31_aaga(x1, x2, x3, x4, x5)  =  U31_aaga(x1, x3, x5)
pS_in_agga(x1, x2, x3, x4)  =  pS_in_agga(x2, x3)
U32_agga(x1, x2, x3, x4, x5)  =  U32_agga(x2, x3, x5)
U33_agga(x1, x2, x3, x4, x5)  =  U33_agga(x1, x2, x3, x5)
pS_out_agga(x1, x2, x3, x4)  =  pS_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x3, x4, x6, x8)
pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
QSH_IN_GA(x1, x2)  =  QSH_IN_GA(x1)
PI_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PI_IN_GGAAAAA(x1, x2)
U20_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_GGAAAAA(x1, x2, x8)
PP_IN_GAGAGA(x1, x2, x3, x4, x5, x6)  =  PP_IN_GAGAGA(x1, x3, x5)
U22_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U22_GAGAGA(x1, x3, x5, x7)
PQ_IN_GAGGA(x1, x2, x3, x4, x5)  =  PQ_IN_GAGGA(x1, x3, x4)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

PI_IN_GGAAAAA(T99, T100, T104, T105, X190, X191, X192) → U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_GGAAAAA(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → PP_IN_GAGAGA(T104, X190, T105, X191, T99, X192)
PP_IN_GAGAGA(T104, T109, T105, X191, T99, X192) → U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_GAGAGA(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → PQ_IN_GAGGA(T105, X191, T109, T99, X192)
PQ_IN_GAGGA(T105, T110, T109, T99, X192) → QSH_IN_GA(T105, T110)
QSH_IN_GA(.(T99, T100), X192) → PI_IN_GGAAAAA(T99, T100, X188, X189, X190, X191, X192)
PP_IN_GAGAGA(T104, T109, T105, X191, T99, X192) → QSH_IN_GA(T104, T109)

The TRS R consists of the following rules:

partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
QSH_IN_GA(x1, x2)  =  QSH_IN_GA(x1)
PI_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PI_IN_GGAAAAA(x1, x2)
U20_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_GGAAAAA(x1, x2, x8)
PP_IN_GAGAGA(x1, x2, x3, x4, x5, x6)  =  PP_IN_GAGAGA(x1, x3, x5)
U22_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U22_GAGAGA(x1, x3, x5, x7)
PQ_IN_GAGGA(x1, x2, x3, x4, x5)  =  PQ_IN_GAGGA(x1, x3, x4)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

PI_IN_GGAAAAA(T99, T100) → U20_GGAAAAA(T99, T100, partF_in_ggaa(T99, T100))
U20_GGAAAAA(T99, T100, partF_out_ggaa(T99, T100, T104, T105)) → PP_IN_GAGAGA(T104, T105, T99)
PP_IN_GAGAGA(T104, T105, T99) → U22_GAGAGA(T104, T105, T99, qsH_in_ga(T104))
U22_GAGAGA(T104, T105, T99, qsH_out_ga(T104, T109)) → PQ_IN_GAGGA(T105, T109, T99)
PQ_IN_GAGGA(T105, T109, T99) → QSH_IN_GA(T105)
QSH_IN_GA(.(T99, T100)) → PI_IN_GGAAAAA(T99, T100)
PP_IN_GAGAGA(T104, T105, T99) → QSH_IN_GA(T104)

The TRS R consists of the following rules:

partF_in_ggaa(T62, .(T63, T64)) → U5_ggaa(T62, T63, T64, pG_in_gggaa(T62, T63, T64))
partF_in_ggaa(T84, .(T85, T86)) → U6_ggaa(T84, T85, T86, partF_in_ggaa(T84, T86))
partF_in_ggaa(T92, []) → partF_out_ggaa(T92, [], [], [])
qsH_in_ga([]) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100)) → U7_ga(T99, T100, pI_in_ggaaaaa(T99, T100))
U5_ggaa(T62, T63, T64, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U6_ggaa(T84, T85, T86, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U7_ga(T99, T100, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
pG_in_gggaa(T62, T63, T64) → U16_gggaa(T62, T63, T64, lessE_in_gg(T62, T63))
pI_in_ggaaaaa(T99, T100) → U20_ggaaaaa(T99, T100, partF_in_ggaa(T99, T100))
U16_gggaa(T62, T63, T64, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, partF_in_ggaa(T62, T64))
U20_ggaaaaa(T99, T100, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, pP_in_gagaga(T104, T105, T99))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U17_gggaa(T62, T63, T64, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U21_ggaaaaa(T99, T100, T104, T105, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
pP_in_gagaga(T104, T105, T99) → U22_gagaga(T104, T105, T99, qsH_in_ga(T104))
U22_gagaga(T104, T105, T99, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, T99, pQ_in_gagga(T105, T109, T99))
U23_gagaga(T104, T109, T105, T99, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
pQ_in_gagga(T105, T109, T99) → U24_gagga(T105, T109, T99, qsH_in_ga(T105))
U24_gagga(T105, T109, T99, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, appJ_in_ggga(T109, T99, T110))
U25_gagga(T105, T110, T109, T99, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
appJ_in_ggga([], T123, T124) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136) → U8_ggga(T133, T134, T135, T136, appJ_in_ggga(T134, T135, T136))
U8_ggga(T133, T134, T135, T136, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))

The set Q consists of the following terms:

partF_in_ggaa(x0, x1)
qsH_in_ga(x0)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3)
U7_ga(x0, x1, x2)
pG_in_gggaa(x0, x1, x2)
pI_in_ggaaaaa(x0, x1)
U16_gggaa(x0, x1, x2, x3)
U20_ggaaaaa(x0, x1, x2)
lessE_in_gg(x0, x1)
U17_gggaa(x0, x1, x2, x3)
U21_ggaaaaa(x0, x1, x2, x3, x4)
U4_gg(x0, x1, x2)
pP_in_gagaga(x0, x1, x2)
U22_gagaga(x0, x1, x2, x3)
U23_gagaga(x0, x1, x2, x3, x4)
pQ_in_gagga(x0, x1, x2)
U24_gagga(x0, x1, x2, x3)
U25_gagga(x0, x1, x2, x3, x4)
appJ_in_ggga(x0, x1, x2)
U8_ggga(x0, x1, x2, x3, x4)

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

(40) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PP_IN_GAGAGA(T104, T105, T99) → U22_GAGAGA(T104, T105, T99, qsH_in_ga(T104))
PP_IN_GAGAGA(T104, T105, T99) → QSH_IN_GA(T104)

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(0) = 0   
POL(PI_IN_GGAAAAA(x₁, x₂)) = 1 + x₂   
POL(PP_IN_GAGAGA(x₁, x₂, x₃)) = 1 + x₁ + x₂   
POL(PQ_IN_GAGGA(x₁, x₂, x₃)) = x₁   
POL(QSH_IN_GA(x₁)) = x₁   
POL(U16_gggaa(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(U17_gggaa(x₁, x₂, x₃, x₄)) = x₄   
POL(U20_GGAAAAA(x₁, x₂, x₃)) = x₃   
POL(U20_ggaaaaa(x₁, x₂, x₃)) = 0   
POL(U21_ggaaaaa(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U22_GAGAGA(x₁, x₂, x₃, x₄)) = x₂   
POL(U22_gagaga(x₁, x₂, x₃, x₄)) = 0   
POL(U23_gagaga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U24_gagga(x₁, x₂, x₃, x₄)) = 0   
POL(U25_gagga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U4_gg(x₁, x₂, x₃)) = 1   
POL(U5_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U6_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U7_ga(x₁, x₂, x₃)) = 0   
POL(U8_ggga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL([]) = 0   
POL(appJ_in_ggga(x₁, x₂, x₃)) = 0   
POL(appJ_out_ggga(x₁, x₂, x₃, x₄)) = 0   
POL(lessE_in_gg(x₁, x₂)) = 1   
POL(lessE_out_gg(x₁, x₂)) = 1   
POL(pG_in_gggaa(x₁, x₂, x₃)) = 1 + x₃   
POL(pG_out_gggaa(x₁, x₂, x₃, x₄, x₅)) = 1 + x₄ + x₅   
POL(pI_in_ggaaaaa(x₁, x₂)) = 0   
POL(pI_out_ggaaaaa(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 0   
POL(pP_in_gagaga(x₁, x₂, x₃)) = 0   
POL(pP_out_gagaga(x₁, x₂, x₃, x₄, x₅, x₆)) = 0   
POL(pQ_in_gagga(x₁, x₂, x₃)) = 0   
POL(pQ_out_gagga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(partF_in_ggaa(x₁, x₂)) = 1 + x₂   
POL(partF_out_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄   
POL(qsH_in_ga(x₁)) = 0   
POL(qsH_out_ga(x₁, x₂)) = 0   
POL(s(x₁)) = 0   

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

partF_in_ggaa(T62, .(T63, T64)) → U5_ggaa(T62, T63, T64, pG_in_gggaa(T62, T63, T64))
partF_in_ggaa(T84, .(T85, T86)) → U6_ggaa(T84, T85, T86, partF_in_ggaa(T84, T86))
partF_in_ggaa(T92, []) → partF_out_ggaa(T92, [], [], [])
pG_in_gggaa(T62, T63, T64) → U16_gggaa(T62, T63, T64, lessE_in_gg(T62, T63))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U16_gggaa(T62, T63, T64, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, partF_in_ggaa(T62, T64))
U17_gggaa(T62, T63, T64, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U6_ggaa(T84, T85, T86, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))

(41) Obligation:

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

PI_IN_GGAAAAA(T99, T100) → U20_GGAAAAA(T99, T100, partF_in_ggaa(T99, T100))
U20_GGAAAAA(T99, T100, partF_out_ggaa(T99, T100, T104, T105)) → PP_IN_GAGAGA(T104, T105, T99)
U22_GAGAGA(T104, T105, T99, qsH_out_ga(T104, T109)) → PQ_IN_GAGGA(T105, T109, T99)
PQ_IN_GAGGA(T105, T109, T99) → QSH_IN_GA(T105)
QSH_IN_GA(.(T99, T100)) → PI_IN_GGAAAAA(T99, T100)

The TRS R consists of the following rules:

partF_in_ggaa(T62, .(T63, T64)) → U5_ggaa(T62, T63, T64, pG_in_gggaa(T62, T63, T64))
partF_in_ggaa(T84, .(T85, T86)) → U6_ggaa(T84, T85, T86, partF_in_ggaa(T84, T86))
partF_in_ggaa(T92, []) → partF_out_ggaa(T92, [], [], [])
qsH_in_ga([]) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100)) → U7_ga(T99, T100, pI_in_ggaaaaa(T99, T100))
U5_ggaa(T62, T63, T64, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U6_ggaa(T84, T85, T86, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U7_ga(T99, T100, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
pG_in_gggaa(T62, T63, T64) → U16_gggaa(T62, T63, T64, lessE_in_gg(T62, T63))
pI_in_ggaaaaa(T99, T100) → U20_ggaaaaa(T99, T100, partF_in_ggaa(T99, T100))
U16_gggaa(T62, T63, T64, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, partF_in_ggaa(T62, T64))
U20_ggaaaaa(T99, T100, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, pP_in_gagaga(T104, T105, T99))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U17_gggaa(T62, T63, T64, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U21_ggaaaaa(T99, T100, T104, T105, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
pP_in_gagaga(T104, T105, T99) → U22_gagaga(T104, T105, T99, qsH_in_ga(T104))
U22_gagaga(T104, T105, T99, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, T99, pQ_in_gagga(T105, T109, T99))
U23_gagaga(T104, T109, T105, T99, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
pQ_in_gagga(T105, T109, T99) → U24_gagga(T105, T109, T99, qsH_in_ga(T105))
U24_gagga(T105, T109, T99, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, appJ_in_ggga(T109, T99, T110))
U25_gagga(T105, T110, T109, T99, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
appJ_in_ggga([], T123, T124) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136) → U8_ggga(T133, T134, T135, T136, appJ_in_ggga(T134, T135, T136))
U8_ggga(T133, T134, T135, T136, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))

The set Q consists of the following terms:

partF_in_ggaa(x0, x1)
qsH_in_ga(x0)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3)
U7_ga(x0, x1, x2)
pG_in_gggaa(x0, x1, x2)
pI_in_ggaaaaa(x0, x1)
U16_gggaa(x0, x1, x2, x3)
U20_ggaaaaa(x0, x1, x2)
lessE_in_gg(x0, x1)
U17_gggaa(x0, x1, x2, x3)
U21_ggaaaaa(x0, x1, x2, x3, x4)
U4_gg(x0, x1, x2)
pP_in_gagaga(x0, x1, x2)
U22_gagaga(x0, x1, x2, x3)
U23_gagaga(x0, x1, x2, x3, x4)
pQ_in_gagga(x0, x1, x2)
U24_gagga(x0, x1, x2, x3)
U25_gagga(x0, x1, x2, x3, x4)
appJ_in_ggga(x0, x1, x2)
U8_ggga(x0, x1, x2, x3, x4)

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

(42) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.

(43) TRUE

(44) Obligation:

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

QSA_IN_GA(.(T22, .(T23, T24)), T9) → PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9)
PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9) → U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → PM_IN_GGAAGAAA(T22, T24, X49, X50, T23, X14, X15, T9)
PM_IN_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9) → U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → PN_IN_GGAGAGA(T23, T45, X14, T46, X15, T22, T9)
PN_IN_GGAGAGA(T23, T45, T93, T46, X15, T22, T9) → QSA_IN_GA(.(T23, T45), T93)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9))
pB_in_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9) → U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U10_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_in_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9))
pM_in_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9) → U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U12_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_in_ggagaga(T23, T45, X14, T46, X15, T22, T9))
pN_in_ggagaga(T23, T45, T93, T46, X15, T22, T9) → U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_in_ga(.(T23, T45), T93))
qsA_in_ga(.(T187, .(T188, T189)), T9) → U2_ga(T187, T188, T189, T9, pC_in_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9))
pC_in_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9) → U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_in_ggaa(T187, T189, T193, T194))
U26_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, partF_out_ggaa(T187, T189, T193, T194)) → U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_in_gaggaga(T193, X14, T188, T194, X15, T187, T9))
pR_in_gaggaga(T193, T198, T188, T194, X15, T187, T9) → U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_in_ga(T193, T198))
qsH_in_ga([], []) → qsH_out_ga([], [])
qsH_in_ga(.(T99, T100), X192) → U7_ga(T99, T100, X192, pI_in_ggaaaaa(T99, T100, X188, X189, X190, X191, X192))
pI_in_ggaaaaa(T99, T100, T104, T105, X190, X191, X192) → U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_in_ggaa(T99, T100, T104, T105))
U20_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, partF_out_ggaa(T99, T100, T104, T105)) → U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_in_gagaga(T104, X190, T105, X191, T99, X192))
pP_in_gagaga(T104, T109, T105, X191, T99, X192) → U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_in_ga(T104, T109))
U22_gagaga(T104, T109, T105, X191, T99, X192, qsH_out_ga(T104, T109)) → U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_in_gagga(T105, X191, T109, T99, X192))
pQ_in_gagga(T105, T110, T109, T99, X192) → U24_gagga(T105, T110, T109, T99, X192, qsH_in_ga(T105, T110))
U24_gagga(T105, T110, T109, T99, X192, qsH_out_ga(T105, T110)) → U25_gagga(T105, T110, T109, T99, X192, appJ_in_ggga(T109, T99, T110, X192))
appJ_in_ggga([], T123, T124, .(T123, T124)) → appJ_out_ggga([], T123, T124, .(T123, T124))
appJ_in_ggga(.(T133, T134), T135, T136, .(T133, X228)) → U8_ggga(T133, T134, T135, T136, X228, appJ_in_ggga(T134, T135, T136, X228))
U8_ggga(T133, T134, T135, T136, X228, appJ_out_ggga(T134, T135, T136, X228)) → appJ_out_ggga(.(T133, T134), T135, T136, .(T133, X228))
U25_gagga(T105, T110, T109, T99, X192, appJ_out_ggga(T109, T99, T110, X192)) → pQ_out_gagga(T105, T110, T109, T99, X192)
U23_gagaga(T104, T109, T105, X191, T99, X192, pQ_out_gagga(T105, X191, T109, T99, X192)) → pP_out_gagaga(T104, T109, T105, X191, T99, X192)
U21_ggaaaaa(T99, T100, T104, T105, X190, X191, X192, pP_out_gagaga(T104, X190, T105, X191, T99, X192)) → pI_out_ggaaaaa(T99, T100, T104, T105, X190, X191, X192)
U7_ga(T99, T100, X192, pI_out_ggaaaaa(T99, T100, X188, X189, X190, X191, X192)) → qsH_out_ga(.(T99, T100), X192)
U28_gaggaga(T193, T198, T188, T194, X15, T187, T9, qsH_out_ga(T193, T198)) → U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_in_gagga(.(T188, T194), X15, T198, T187, T9))
pO_in_gagga(T46, T94, T93, T22, T9) → U18_gagga(T46, T94, T93, T22, T9, qsH_in_ga(T46, T94))
U18_gagga(T46, T94, T93, T22, T9, qsH_out_ga(T46, T94)) → U19_gagga(T46, T94, T93, T22, T9, appK_in_ggga(T93, T22, T94, T9))
appK_in_ggga([], T153, T154, .(T153, T154)) → appK_out_ggga([], T153, T154, .(T153, T154))
appK_in_ggga(.(T165, T166), T167, T168, .(T165, T170)) → U9_ggga(T165, T166, T167, T168, T170, appK_in_ggga(T166, T167, T168, T170))
U9_ggga(T165, T166, T167, T168, T170, appK_out_ggga(T166, T167, T168, T170)) → appK_out_ggga(.(T165, T166), T167, T168, .(T165, T170))
U19_gagga(T46, T94, T93, T22, T9, appK_out_ggga(T93, T22, T94, T9)) → pO_out_gagga(T46, T94, T93, T22, T9)
U29_gaggaga(T193, T198, T188, T194, X15, T187, T9, pO_out_gagga(.(T188, T194), X15, T198, T187, T9)) → pR_out_gaggaga(T193, T198, T188, T194, X15, T187, T9)
U27_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9, pR_out_gaggaga(T193, X14, T188, T194, X15, T187, T9)) → pC_out_ggaaagaa(T187, T189, T193, T194, X14, T188, X15, T9)
U2_ga(T187, T188, T189, T9, pC_out_ggaaagaa(T187, T189, X285, X286, X14, T188, X15, T9)) → qsA_out_ga(.(T187, .(T188, T189)), T9)
qsA_in_ga(.(T201, []), T9) → U3_ga(T201, T9, pD_in_aaga(X14, X15, T201, T9))
pD_in_aaga(T206, X15, T201, T9) → U30_aaga(T206, X15, T201, T9, qsL_in_a(T206))
qsL_in_a([]) → qsL_out_a([])
U30_aaga(T206, X15, T201, T9, qsL_out_a(T206)) → U31_aaga(T206, X15, T201, T9, pS_in_agga(X15, T206, T201, T9))
pS_in_agga(T211, T206, T201, T9) → U32_agga(T211, T206, T201, T9, qsL_in_a(T211))
U32_agga(T211, T206, T201, T9, qsL_out_a(T211)) → U33_agga(T211, T206, T201, T9, appK_in_ggga(T206, T201, T211, T9))
U33_agga(T211, T206, T201, T9, appK_out_ggga(T206, T201, T211, T9)) → pS_out_agga(T211, T206, T201, T9)
U31_aaga(T206, X15, T201, T9, pS_out_agga(X15, T206, T201, T9)) → pD_out_aaga(T206, X15, T201, T9)
U3_ga(T201, T9, pD_out_aaga(X14, X15, T201, T9)) → qsA_out_ga(.(T201, []), T9)
U14_ggagaga(T23, T45, T93, T46, X15, T22, T9, qsA_out_ga(.(T23, T45), T93)) → U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_in_gagga(T46, X15, T93, T22, T9))
U15_ggagaga(T23, T45, T93, T46, X15, T22, T9, pO_out_gagga(T46, X15, T93, T22, T9)) → pN_out_ggagaga(T23, T45, T93, T46, X15, T22, T9)
U13_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9, pN_out_ggagaga(T23, T45, X14, T46, X15, T22, T9)) → pM_out_ggaagaaa(T22, T24, T45, T46, T23, X14, X15, T9)
U11_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9, pM_out_ggaagaaa(T22, T24, X49, X50, T23, X14, X15, T9)) → pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X49, X50, X14, X15, T9)) → qsA_out_ga(.(T22, .(T23, T24)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2)  =  qsA_in_ga(x1)
[]  =  []
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
pB_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_in_gggaaaaa(x1, x2, x3)
U10_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_gggaaaaa(x1, x2, x3, x9)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
pM_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_in_ggaagaaa(x1, x2, x5)
U12_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_ggaagaaa(x1, x2, x5, x9)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_ggaagaaa(x1, x2, x3, x4, x5, x9)
pN_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_in_ggagaga(x1, x2, x4, x6)
U14_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_ggaaagaa(x1, x2, x6)
U26_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_ggaaagaa(x1, x2, x6, x9)
U27_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U27_ggaaagaa(x1, x2, x3, x4, x6, x9)
pR_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_in_gaggaga(x1, x3, x4, x6)
U28_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U28_gaggaga(x1, x3, x4, x6, x8)
qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)
qsH_out_ga(x1, x2)  =  qsH_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x2, x4)
pI_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_in_ggaaaaa(x1, x2)
U20_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U20_ggaaaaa(x1, x2, x8)
U21_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21_ggaaaaa(x1, x2, x3, x4, x8)
pP_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_in_gagaga(x1, x3, x5)
U22_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U22_gagaga(x1, x3, x5, x7)
U23_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U23_gagaga(x1, x2, x3, x5, x7)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U24_gagga(x1, x2, x3, x4, x5, x6)  =  U24_gagga(x1, x3, x4, x6)
U25_gagga(x1, x2, x3, x4, x5, x6)  =  U25_gagga(x1, x2, x3, x4, x6)
appJ_in_ggga(x1, x2, x3, x4)  =  appJ_in_ggga(x1, x2, x3)
appJ_out_ggga(x1, x2, x3, x4)  =  appJ_out_ggga(x1, x2, x3, x4)
U8_ggga(x1, x2, x3, x4, x5, x6)  =  U8_ggga(x1, x2, x3, x4, x6)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pP_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pP_out_gagaga(x1, x2, x3, x4, x5, x6)
pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pI_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U29_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_gaggaga(x1, x2, x3, x4, x6, x8)
pO_in_gagga(x1, x2, x3, x4, x5)  =  pO_in_gagga(x1, x3, x4)
U18_gagga(x1, x2, x3, x4, x5, x6)  =  U18_gagga(x1, x3, x4, x6)
U19_gagga(x1, x2, x3, x4, x5, x6)  =  U19_gagga(x1, x2, x3, x4, x6)
appK_in_ggga(x1, x2, x3, x4)  =  appK_in_ggga(x1, x2, x3)
appK_out_ggga(x1, x2, x3, x4)  =  appK_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
pO_out_gagga(x1, x2, x3, x4, x5)  =  pO_out_gagga(x1, x2, x3, x4, x5)
pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pR_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U30_aaga(x1, x2, x3, x4, x5)  =  U30_aaga(x3, x5)
qsL_in_a(x1)  =  qsL_in_a
qsL_out_a(x1)  =  qsL_out_a(x1)
U31_aaga(x1, x2, x3, x4, x5)  =  U31_aaga(x1, x3, x5)
pS_in_agga(x1, x2, x3, x4)  =  pS_in_agga(x2, x3)
U32_agga(x1, x2, x3, x4, x5)  =  U32_agga(x2, x3, x5)
U33_agga(x1, x2, x3, x4, x5)  =  U33_agga(x1, x2, x3, x5)
pS_out_agga(x1, x2, x3, x4)  =  pS_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x3, x4, x6, x8)
pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pN_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pM_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)
pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pB_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
QSA_IN_GA(x1, x2)  =  QSA_IN_GA(x1)
PB_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGGAAAAA(x1, x2, x3)
U10_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_GGGAAAAA(x1, x2, x3, x9)
PM_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PM_IN_GGAAGAAA(x1, x2, x5)
U12_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_GGAAGAAA(x1, x2, x5, x9)
PN_IN_GGAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PN_IN_GGAGAGA(x1, x2, x4, x6)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

QSA_IN_GA(.(T22, .(T23, T24)), T9) → PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9)
PB_IN_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9) → U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_in_gg(T22, T23))
U10_GGGAAAAA(T22, T23, T24, X49, X50, X14, X15, T9, lessE_out_gg(T22, T23)) → PM_IN_GGAAGAAA(T22, T24, X49, X50, T23, X14, X15, T9)
PM_IN_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9) → U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_in_ggaa(T22, T24, T45, T46))
U12_GGAAGAAA(T22, T24, T45, T46, T23, X14, X15, T9, partF_out_ggaa(T22, T24, T45, T46)) → PN_IN_GGAGAGA(T23, T45, X14, T46, X15, T22, T9)
PN_IN_GGAGAGA(T23, T45, T93, T46, X15, T22, T9) → QSA_IN_GA(.(T23, T45), T93)

The TRS R consists of the following rules:

lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
partF_in_ggaa(T62, .(T63, T64), .(T63, X115), X116) → U5_ggaa(T62, T63, T64, X115, X116, pG_in_gggaa(T62, T63, T64, X115, X116))
partF_in_ggaa(T84, .(T85, T86), X161, .(T85, X162)) → U6_ggaa(T84, T85, T86, X161, X162, partF_in_ggaa(T84, T86, X161, X162))
partF_in_ggaa(T92, [], [], []) → partF_out_ggaa(T92, [], [], [])
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U5_ggaa(T62, T63, T64, X115, X116, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U6_ggaa(T84, T85, T86, X161, X162, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
pG_in_gggaa(T62, T63, T64, X115, X116) → U16_gggaa(T62, T63, T64, X115, X116, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, X115, X116, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, X115, X116, partF_in_ggaa(T62, T64, X115, X116))
U17_gggaa(T62, T63, T64, X115, X116, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
lessE_in_gg(x1, x2)  =  lessE_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
lessE_out_gg(x1, x2)  =  lessE_out_gg(x1, x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
partF_in_ggaa(x1, x2, x3, x4)  =  partF_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
pG_in_gggaa(x1, x2, x3, x4, x5)  =  pG_in_gggaa(x1, x2, x3)
U16_gggaa(x1, x2, x3, x4, x5, x6)  =  U16_gggaa(x1, x2, x3, x6)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
QSA_IN_GA(x1, x2)  =  QSA_IN_GA(x1)
PB_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PB_IN_GGGAAAAA(x1, x2, x3)
U10_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U10_GGGAAAAA(x1, x2, x3, x9)
PM_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PM_IN_GGAAGAAA(x1, x2, x5)
U12_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_GGAAGAAA(x1, x2, x5, x9)
PN_IN_GGAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PN_IN_GGAGAGA(x1, x2, x4, x6)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

QSA_IN_GA(.(T22, .(T23, T24))) → PB_IN_GGGAAAAA(T22, T23, T24)
PB_IN_GGGAAAAA(T22, T23, T24) → U10_GGGAAAAA(T22, T23, T24, lessE_in_gg(T22, T23))
U10_GGGAAAAA(T22, T23, T24, lessE_out_gg(T22, T23)) → PM_IN_GGAAGAAA(T22, T24, T23)
PM_IN_GGAAGAAA(T22, T24, T23) → U12_GGAAGAAA(T22, T24, T23, partF_in_ggaa(T22, T24))
U12_GGAAGAAA(T22, T24, T23, partF_out_ggaa(T22, T24, T45, T46)) → PN_IN_GGAGAGA(T23, T45, T46, T22)
PN_IN_GGAGAGA(T23, T45, T46, T22) → QSA_IN_GA(.(T23, T45))

The TRS R consists of the following rules:

lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
partF_in_ggaa(T62, .(T63, T64)) → U5_ggaa(T62, T63, T64, pG_in_gggaa(T62, T63, T64))
partF_in_ggaa(T84, .(T85, T86)) → U6_ggaa(T84, T85, T86, partF_in_ggaa(T84, T86))
partF_in_ggaa(T92, []) → partF_out_ggaa(T92, [], [], [])
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U5_ggaa(T62, T63, T64, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U6_ggaa(T84, T85, T86, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
pG_in_gggaa(T62, T63, T64) → U16_gggaa(T62, T63, T64, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, partF_in_ggaa(T62, T64))
U17_gggaa(T62, T63, T64, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)

The set Q consists of the following terms:

lessE_in_gg(x0, x1)
partF_in_ggaa(x0, x1)
U4_gg(x0, x1, x2)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3)
pG_in_gggaa(x0, x1, x2)
U16_gggaa(x0, x1, x2, x3)
U17_gggaa(x0, x1, x2, x3)

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

(49) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

QSA_IN_GA(.(T22, .(T23, T24))) → PB_IN_GGGAAAAA(T22, T23, T24)

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(0) = 0   
POL(PB_IN_GGGAAAAA(x₁, x₂, x₃)) = 1 + x₃   
POL(PM_IN_GGAAGAAA(x₁, x₂, x₃)) = 1 + x₂   
POL(PN_IN_GGAGAGA(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(QSA_IN_GA(x₁)) = x₁   
POL(U10_GGGAAAAA(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U12_GGAAGAAA(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U16_gggaa(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U17_gggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U4_gg(x₁, x₂, x₃)) = 0   
POL(U5_ggaa(x₁, x₂, x₃, x₄)) = x₄   
POL(U6_ggaa(x₁, x₂, x₃, x₄)) = x₄   
POL([]) = 0   
POL(lessE_in_gg(x₁, x₂)) = 0   
POL(lessE_out_gg(x₁, x₂)) = 0   
POL(pG_in_gggaa(x₁, x₂, x₃)) = 1 + x₃   
POL(pG_out_gggaa(x₁, x₂, x₃, x₄, x₅)) = 1 + x₄   
POL(partF_in_ggaa(x₁, x₂)) = x₂   
POL(partF_out_ggaa(x₁, x₂, x₃, x₄)) = x₃   
POL(s(x₁)) = 0   

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

partF_in_ggaa(T62, .(T63, T64)) → U5_ggaa(T62, T63, T64, pG_in_gggaa(T62, T63, T64))
partF_in_ggaa(T84, .(T85, T86)) → U6_ggaa(T84, T85, T86, partF_in_ggaa(T84, T86))
partF_in_ggaa(T92, []) → partF_out_ggaa(T92, [], [], [])
U16_gggaa(T62, T63, T64, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, partF_in_ggaa(T62, T64))
U17_gggaa(T62, T63, T64, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)
U5_ggaa(T62, T63, T64, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
pG_in_gggaa(T62, T63, T64) → U16_gggaa(T62, T63, T64, lessE_in_gg(T62, T63))
U6_ggaa(T84, T85, T86, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))

(50) Obligation:

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

PB_IN_GGGAAAAA(T22, T23, T24) → U10_GGGAAAAA(T22, T23, T24, lessE_in_gg(T22, T23))
U10_GGGAAAAA(T22, T23, T24, lessE_out_gg(T22, T23)) → PM_IN_GGAAGAAA(T22, T24, T23)
PM_IN_GGAAGAAA(T22, T24, T23) → U12_GGAAGAAA(T22, T24, T23, partF_in_ggaa(T22, T24))
U12_GGAAGAAA(T22, T24, T23, partF_out_ggaa(T22, T24, T45, T46)) → PN_IN_GGAGAGA(T23, T45, T46, T22)
PN_IN_GGAGAGA(T23, T45, T46, T22) → QSA_IN_GA(.(T23, T45))

The TRS R consists of the following rules:

lessE_in_gg(0, s(T33)) → lessE_out_gg(0, s(T33))
lessE_in_gg(s(T38), s(T39)) → U4_gg(T38, T39, lessE_in_gg(T38, T39))
partF_in_ggaa(T62, .(T63, T64)) → U5_ggaa(T62, T63, T64, pG_in_gggaa(T62, T63, T64))
partF_in_ggaa(T84, .(T85, T86)) → U6_ggaa(T84, T85, T86, partF_in_ggaa(T84, T86))
partF_in_ggaa(T92, []) → partF_out_ggaa(T92, [], [], [])
U4_gg(T38, T39, lessE_out_gg(T38, T39)) → lessE_out_gg(s(T38), s(T39))
U5_ggaa(T62, T63, T64, pG_out_gggaa(T62, T63, T64, X115, X116)) → partF_out_ggaa(T62, .(T63, T64), .(T63, X115), X116)
U6_ggaa(T84, T85, T86, partF_out_ggaa(T84, T86, X161, X162)) → partF_out_ggaa(T84, .(T85, T86), X161, .(T85, X162))
pG_in_gggaa(T62, T63, T64) → U16_gggaa(T62, T63, T64, lessE_in_gg(T62, T63))
U16_gggaa(T62, T63, T64, lessE_out_gg(T62, T63)) → U17_gggaa(T62, T63, T64, partF_in_ggaa(T62, T64))
U17_gggaa(T62, T63, T64, partF_out_ggaa(T62, T64, X115, X116)) → pG_out_gggaa(T62, T63, T64, X115, X116)

The set Q consists of the following terms:

lessE_in_gg(x0, x1)
partF_in_ggaa(x0, x1)
U4_gg(x0, x1, x2)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3)
pG_in_gggaa(x0, x1, x2)
U16_gggaa(x0, x1, x2, x3)
U17_gggaa(x0, x1, x2, x3)

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

(51) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.

(52) TRUE