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

(0) Obligation:

Clauses:

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

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];
9 [label="9: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))) | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX4 is free\nX5 is free\nX6 is free\nqs(T1, T2) !=? qs(.(X4, X5), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 2 | ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 3 | ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="B: ','(','(gt(T22, T23), part(T22, T24, X44, X45)), ','(qs(.(T23, X44), X9), ','(qs(X45, X10), app(X9, .(T22, X10), T9))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX9 is free\nX10 is free\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="E: gt(T22, T23)\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="O: ','(part(T22, T24, X44, X45), ','(qs(.(T23, X44), X9), ','(qs(X45, X10), app(X9, .(T22, X10), T9))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX9 is free\nX10 is free\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: gt(T22, T23), SCOPE: 3, CLAUSE: 7 | gt(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: gt(T22, T23), SCOPE: 3, CLAUSE: 7\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: gt(T22, T23), SCOPE: 3, CLAUSE: 8\n\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: gt(T37, T38)\n\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="F: part(T22, T24, X44, X45)\n\nT22 is ground\nT24 is ground\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="P: ','(qs(.(T23, T44), X9), ','(qs(T45, X10), app(X9, .(T22, X10), T9)))\n\nT22 is ground\nT23 is ground\nT44 is ground\nT45 is ground\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: part(T22, T24, X44, X45), SCOPE: 4, CLAUSE: 2 | part(T22, T24, X44, X45), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X44, X45), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: part(T22, T24, X44, X45), SCOPE: 4, CLAUSE: 2\n\nT22 is ground\nT24 is ground\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: part(T22, T24, X44, X45), SCOPE: 4, CLAUSE: 3 | part(T22, T24, X44, X45), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="G: ','(gt(T61, T62), part(T61, T63, X109, X110))\n\nT61 is ground\nT62 is ground\nT63 is ground\nX109 is free\nX110 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: gt(T61, T62)\n\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: part(T61, T63, X109, X110)\n\nT61 is ground\nT63 is ground\nX109 is free\nX110 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: part(T22, T24, X44, X45), SCOPE: 4, CLAUSE: 3\n\nT22 is ground\nT24 is ground\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: part(T22, T24, X44, X45), SCOPE: 4, CLAUSE: 4\n\nT22 is ground\nT24 is ground\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="H: ','(le(T83, T84), part(T83, T85, X155, X156))\n\nT83 is ground\nT84 is ground\nT85 is ground\nX155 is free\nX156 is free", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="I: le(T83, T84)\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: part(T83, T85, X155, X156)\n\nT83 is ground\nT85 is ground\nX155 is free\nX156 is free", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: le(T83, T84), SCOPE: 5, CLAUSE: 9 | le(T83, T84), SCOPE: 5, CLAUSE: 10 | le(T83, T84), SCOPE: 5, CLAUSE: 11\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: le(T83, T84), SCOPE: 5, CLAUSE: 9\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: le(T83, T84), SCOPE: 5, CLAUSE: 10 | le(T83, T84), SCOPE: 5, CLAUSE: 11\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: le(T98, T99)\n\nT98 is ground\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: le(T83, T84), SCOPE: 5, CLAUSE: 10\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: le(T83, T84), SCOPE: 5, CLAUSE: 11\n\nT83 is ground\nT84 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: qs(.(T23, T44), X9)\n\nT23 is ground\nT44 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="Q: ','(qs(T45, X10), app(T110, .(T22, X10), T9))\n\nT22 is ground\nT45 is ground\nT110 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="J: qs(T45, X10)\n\nT45 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="M: app(T110, .(T22, T115), T9)\n\nT22 is ground\nT110 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: qs(T45, X10), SCOPE: 6, CLAUSE: 0 | qs(T45, X10), SCOPE: 6, CLAUSE: 1\n\nT45 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: qs(T45, X10), SCOPE: 6, CLAUSE: 0\n\nT45 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: qs(T45, X10), SCOPE: 6, CLAUSE: 1\n\nT45 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="K: ','(part(T126, T127, X242, X243), ','(qs(X242, X244), ','(qs(X243, X245), app(X244, .(T126, X245), X246))))\n\nT126 is ground\nT127 is ground\nX246 is free\nX242 is free\nX243 is free\nX244 is free\nX245 is free", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: part(T126, T127, X242, X243)\n\nT126 is ground\nT127 is ground\nX242 is free\nX243 is free", fontsize=16, style=filled, fillcolor=lightyellow];
167 [label="R: ','(qs(T131, X244), ','(qs(T132, X245), app(X244, .(T126, X245), X246)))\n\nT126 is ground\nT131 is ground\nT132 is ground\nX246 is free\nX244 is free\nX245 is free", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: qs(T131, X244)\n\nT131 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
178 [label="S: ','(qs(T132, X245), app(T138, .(T126, X245), X246))\n\nT126 is ground\nT132 is ground\nT138 is ground\nX246 is free\nX245 is free", fontsize=16, style=filled, fillcolor=lightyellow];
188 [label="188: qs(T132, X245)\n\nT132 is ground\nX245 is free", fontsize=16, style=filled, fillcolor=lightyellow];
189 [label="L: app(T138, .(T126, T143), X246)\n\nT126 is ground\nT138 is ground\nT143 is ground\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
196 [label="196: app(T138, .(T126, T143), X246), SCOPE: 7, CLAUSE: 5 | app(T138, .(T126, T143), X246), SCOPE: 7, CLAUSE: 6\n\nT126 is ground\nT138 is ground\nT143 is ground\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="198: app(T138, .(T126, T143), X246), SCOPE: 7, CLAUSE: 5\n\nT126 is ground\nT138 is ground\nT143 is ground\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: app(T138, .(T126, T143), X246), SCOPE: 7, CLAUSE: 6\n\nT126 is ground\nT138 is ground\nT143 is ground\nX246 is free", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: app(T167, .(T168, T169), X322)\n\nT167 is ground\nT168 is ground\nT169 is ground\nX322 is free", fontsize=16, style=filled, fillcolor=lightyellow];
205 [label="205: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
210 [label="210: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
211 [label="211: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="212: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="213: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="216: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: app(T110, .(T22, T115), T9), SCOPE: 8, CLAUSE: 5 | app(T110, .(T22, T115), T9), SCOPE: 8, CLAUSE: 6\n\nT22 is ground\nT110 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
225 [label="225: app(T110, .(T22, T115), T9), SCOPE: 8, CLAUSE: 5\n\nT22 is ground\nT110 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: app(T110, .(T22, T115), T9), SCOPE: 8, CLAUSE: 6\n\nT22 is ground\nT110 is ground\nT115 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="228: app(T206, .(T207, T208), T210)\n\nT206 is ground\nT207 is ground\nT208 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
229 [label="229: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
232 [label="232: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
233 [label="233: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
234 [label="234: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
237 [label="237: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="238: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4 | ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
246 [label="C: ','(','(le(T234, T235), part(T234, T236, X395, X396)), ','(qs(X395, X9), ','(qs(.(T235, X396), X10), app(X9, .(T234, X10), T9))))\n\nT234 is ground\nT235 is ground\nT236 is ground\nX9 is free\nX10 is free\nX395 is free\nX396 is free", fontsize=16, style=filled, fillcolor=lightyellow];
247 [label="247: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
250 [label="250: le(T234, T235)\n\nT234 is ground\nT235 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
251 [label="T: ','(part(T234, T236, X395, X396), ','(qs(X395, X9), ','(qs(.(T235, X396), X10), app(X9, .(T234, X10), T9))))\n\nT234 is ground\nT235 is ground\nT236 is ground\nX9 is free\nX10 is free\nX395 is free\nX396 is free", fontsize=16, style=filled, fillcolor=lightyellow];
257 [label="257: part(T234, T236, X395, X396)\n\nT234 is ground\nT236 is ground\nX395 is free\nX396 is free", fontsize=16, style=filled, fillcolor=lightyellow];
258 [label="U: ','(qs(T244, X9), ','(qs(.(T235, T245), X10), app(X9, .(T234, X10), T9)))\n\nT234 is ground\nT235 is ground\nT244 is ground\nT245 is ground\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
259 [label="259: qs(T244, X9)\n\nT244 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
262 [label="262: ','(qs(.(T235, T245), X10), app(T251, .(T234, X10), T9))\n\nT234 is ground\nT235 is ground\nT245 is ground\nT251 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
267 [label="267: ','(part(T6, T7, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(T6, X10), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX7 is free\nX8 is free\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
268 [label="268: ?_2 | qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
269 [label="D: ','(qs([], X9), ','(qs([], X10), app(X9, .(T260, X10), T9)))\n\nT260 is ground\nX9 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
270 [label="270: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
273 [label="N: qs([], X9)\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
274 [label="V: ','(qs([], X10), app(T261, .(T260, X10), T9))\n\nT260 is ground\nT261 is ground\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
275 [label="275: qs([], X9), SCOPE: 9, CLAUSE: 0 | qs([], X9), SCOPE: 9, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
276 [label="276: qs([], X9), SCOPE: 9, CLAUSE: 1\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
277 [label="277: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
278 [label="278: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
281 [label="281: qs([], X10)\n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
282 [label="282: app(T261, .(T260, T262), T9)\n\nT260 is ground\nT261 is ground\nT262 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
287 [label="287: qs(.(T6, T7), T2), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
288 [label="288: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
289 [label="289: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
290 [label="290: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
291 [label="291: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
292 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 292 [arrowhead = none ];
292 -> 5;

293 [label="EVAL with clause\nqs(.(X4, X5), X6) :- ','(part(X4, X5, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(X4, X10), X6)))).\nand substitutionX4 -> T6,\nX5 -> T7,\nT1 -> .(T6, T7),\nT2 -> T9,\nX6 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 293 [arrowhead = none ];
293 -> 9;

294 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 294 [arrowhead = none ];
294 -> 10;

295 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 295 [arrowhead = none ];
295 -> 20;

296 [label="EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 296 [arrowhead = none ];
296 -> 289;

297 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 297 [arrowhead = none ];
297 -> 290;

298 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 298 [arrowhead = none ];
298 -> 22;

299 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 299 [arrowhead = none ];
299 -> 23;

300 [label="EVAL with clause\npart(X39, .(X40, X41), .(X40, X42), X43) :- ','(gt(X39, X40), part(X39, X41, X42, X43)).\nand substitutionT6 -> T22,\nX39 -> T22,\nX40 -> T23,\nX41 -> T24,\nT7 -> .(T23, T24),\nX42 -> X44,\nX7 -> .(T23, X44),\nX8 -> X45,\nX43 -> X45", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 300 [arrowhead = none ];
300 -> 28;

301 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 301 [arrowhead = none ];
301 -> 31;

302 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 302 [arrowhead = none ];
302 -> 237;

303 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 303 [arrowhead = none ];
303 -> 238;

304 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
28 -> 304 [arrowhead = none ];
304 -> 33;

305 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nT23 is ground", fontsize=14, style = filled, fillcolor = violet];
28 -> 305 [arrowhead = none ];
305 -> 34;

306 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 306 [arrowhead = none ];
306 -> 36;

307 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
34 -> 307 [arrowhead = none ];
307 -> 57;

308 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nT24 is ground\nT44 is ground\nT45 is ground\nreplacements:X44 -> T44,\nX45 -> T45", fontsize=14, style = filled, fillcolor = violet];
34 -> 308 [arrowhead = none ];
308 -> 58;

309 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 309 [arrowhead = none ];
309 -> 39;

310 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 310 [arrowhead = none ];
310 -> 40;

311 [label="EVAL with clause\ngt(s(X58), s(X59)) :- gt(X58, X59).\nand substitutionX58 -> T37,\nT22 -> s(T37),\nX59 -> T38,\nT23 -> s(T38)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 311 [arrowhead = none ];
311 -> 43;

312 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 312 [arrowhead = none ];
312 -> 44;

313 [label="EVAL with clause\ngt(s(0), 0).\nand substitutionT22 -> s(0),\nT23 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 313 [arrowhead = none ];
313 -> 49;

314 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 314 [arrowhead = none ];
314 -> 50;

315 [label="INSTANCE with matching:\nT22 -> T37\nT23 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 315 [arrowhead = none , style=dashed];
315 -> 33[style=dashed];

316 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 316 [arrowhead = none ];
316 -> 51;

317 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 317 [arrowhead = none ];
317 -> 59;

318 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 318 [arrowhead = none ];
318 -> 133;

319 [label="SPLIT 2\nnew knowledge:\nT23 is ground\nT44 is ground\nT110 is ground\nreplacements:X9 -> T110", fontsize=14, style = filled, fillcolor = violet];
58 -> 319 [arrowhead = none ];
319 -> 134;

320 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 320 [arrowhead = none ];
320 -> 64;

321 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
59 -> 321 [arrowhead = none ];
321 -> 65;

322 [label="EVAL with clause\npart(X104, .(X105, X106), .(X105, X107), X108) :- ','(gt(X104, X105), part(X104, X106, X107, X108)).\nand substitutionT22 -> T61,\nX104 -> T61,\nX105 -> T62,\nX106 -> T63,\nT24 -> .(T62, T63),\nX107 -> X109,\nX44 -> .(T62, X109),\nX45 -> X110,\nX108 -> X110", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 322 [arrowhead = none ];
322 -> 72;

323 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 323 [arrowhead = none ];
323 -> 74;

324 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 324 [arrowhead = none ];
324 -> 86;

325 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 325 [arrowhead = none ];
325 -> 87;

326 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
72 -> 326 [arrowhead = none ];
326 -> 77;

327 [label="SPLIT 2\nnew knowledge:\nT61 is ground\nT62 is ground", fontsize=14, style = filled, fillcolor = violet];
72 -> 327 [arrowhead = none ];
327 -> 78;

328 [label="INSTANCE with matching:\nT22 -> T61\nT23 -> T62", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 328 [arrowhead = none , style=dashed];
328 -> 33[style=dashed];

329 [label="INSTANCE with matching:\nT22 -> T61\nT24 -> T63\nX44 -> X109\nX45 -> X110", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 329 [arrowhead = none , style=dashed];
329 -> 57[style=dashed];

330 [label="EVAL with clause\npart(X150, .(X151, X152), X153, .(X151, X154)) :- ','(le(X150, X151), part(X150, X152, X153, X154)).\nand substitutionT22 -> T83,\nX150 -> T83,\nX151 -> T84,\nX152 -> T85,\nT24 -> .(T84, T85),\nX44 -> X155,\nX153 -> X155,\nX154 -> X156,\nX45 -> .(T84, X156)", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 330 [arrowhead = none ];
330 -> 97;

331 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 331 [arrowhead = none ];
331 -> 99;

332 [label="EVAL with clause\npart(X182, [], [], []).\nand substitutionT22 -> T107,\nX182 -> T107,\nT24 -> [],\nX44 -> [],\nX45 -> []", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 332 [arrowhead = none ];
332 -> 128;

333 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 333 [arrowhead = none ];
333 -> 129;

334 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
97 -> 334 [arrowhead = none ];
334 -> 102;

335 [label="SPLIT 2\nnew knowledge:\nT83 is ground\nT84 is ground", fontsize=14, style = filled, fillcolor = violet];
97 -> 335 [arrowhead = none ];
335 -> 103;

336 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
102 -> 336 [arrowhead = none ];
336 -> 104;

337 [label="INSTANCE with matching:\nT22 -> T83\nT24 -> T85\nX44 -> X155\nX45 -> X156", fontsize=14, style = filled, fillcolor = lightgrey];
103 -> 337 [arrowhead = none , style=dashed];
337 -> 57[style=dashed];

338 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 338 [arrowhead = none ];
338 -> 106;

339 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 339 [arrowhead = none ];
339 -> 107;

340 [label="EVAL with clause\nle(s(X169), s(X170)) :- le(X169, X170).\nand substitutionX169 -> T98,\nT83 -> s(T98),\nX170 -> T99,\nT84 -> s(T99)", fontsize=14, style = filled, fillcolor = lightblue];
106 -> 340 [arrowhead = none ];
340 -> 109;

341 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
106 -> 341 [arrowhead = none ];
341 -> 110;

342 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
107 -> 342 [arrowhead = none ];
342 -> 113;

343 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
107 -> 343 [arrowhead = none ];
343 -> 114;

344 [label="INSTANCE with matching:\nT83 -> T98\nT84 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 344 [arrowhead = none , style=dashed];
344 -> 102[style=dashed];

345 [label="EVAL with clause\nle(0, s(0)).\nand substitutionT83 -> 0,\nT84 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 345 [arrowhead = none ];
345 -> 116;

346 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 346 [arrowhead = none ];
346 -> 119;

347 [label="EVAL with clause\nle(0, 0).\nand substitutionT83 -> 0,\nT84 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
114 -> 347 [arrowhead = none ];
347 -> 121;

348 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
114 -> 348 [arrowhead = none ];
348 -> 122;

349 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
116 -> 349 [arrowhead = none ];
349 -> 120;

350 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
121 -> 350 [arrowhead = none ];
350 -> 123;

351 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 351 [arrowhead = none ];
351 -> 130;

352 [label="INSTANCE with matching:\nT1 -> .(T23, T44)\nT2 -> X9", fontsize=14, style = filled, fillcolor = lightgrey];
133 -> 352 [arrowhead = none , style=dashed];
352 -> 3[style=dashed];

353 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
134 -> 353 [arrowhead = none ];
353 -> 139;

354 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nT115 is ground\nreplacements:X10 -> T115", fontsize=14, style = filled, fillcolor = violet];
134 -> 354 [arrowhead = none ];
354 -> 140;

355 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
139 -> 355 [arrowhead = none ];
355 -> 142;

356 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
140 -> 356 [arrowhead = none ];
356 -> 222;

357 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
142 -> 357 [arrowhead = none ];
357 -> 144;

358 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
142 -> 358 [arrowhead = none ];
358 -> 145;

359 [label="EVAL with clause\nqs(.(X239, X240), X241) :- ','(part(X239, X240, X242, X243), ','(qs(X242, X244), ','(qs(X243, X245), app(X244, .(X239, X245), X241)))).\nand substitutionX239 -> T126,\nX240 -> T127,\nT45 -> .(T126, T127),\nX10 -> X246,\nX241 -> X246", fontsize=14, style = filled, fillcolor = lightblue];
144 -> 359 [arrowhead = none ];
359 -> 153;

360 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
144 -> 360 [arrowhead = none ];
360 -> 154;

361 [label="EVAL with clause\nqs([], []).\nand substitutionT45 -> [],\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
145 -> 361 [arrowhead = none ];
361 -> 213;

362 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
145 -> 362 [arrowhead = none ];
362 -> 215;

363 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
153 -> 363 [arrowhead = none ];
363 -> 166;

364 [label="SPLIT 2\nnew knowledge:\nT126 is ground\nT127 is ground\nT131 is ground\nT132 is ground\nreplacements:X242 -> T131,\nX243 -> T132", fontsize=14, style = filled, fillcolor = violet];
153 -> 364 [arrowhead = none ];
364 -> 167;

365 [label="INSTANCE with matching:\nT22 -> T126\nT24 -> T127\nX44 -> X242\nX45 -> X243", fontsize=14, style = filled, fillcolor = lightgrey];
166 -> 365 [arrowhead = none , style=dashed];
365 -> 57[style=dashed];

366 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
167 -> 366 [arrowhead = none ];
366 -> 177;

367 [label="SPLIT 2\nnew knowledge:\nT131 is ground\nT138 is ground\nreplacements:X244 -> T138", fontsize=14, style = filled, fillcolor = violet];
167 -> 367 [arrowhead = none ];
367 -> 178;

368 [label="INSTANCE with matching:\nT45 -> T131\nX10 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
177 -> 368 [arrowhead = none , style=dashed];
368 -> 139[style=dashed];

369 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
178 -> 369 [arrowhead = none ];
369 -> 188;

370 [label="SPLIT 2\nnew knowledge:\nT132 is ground\nT143 is ground\nreplacements:X245 -> T143", fontsize=14, style = filled, fillcolor = violet];
178 -> 370 [arrowhead = none ];
370 -> 189;

371 [label="INSTANCE with matching:\nT45 -> T132\nX10 -> X245", fontsize=14, style = filled, fillcolor = lightgrey];
188 -> 371 [arrowhead = none , style=dashed];
371 -> 139[style=dashed];

372 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
189 -> 372 [arrowhead = none ];
372 -> 196;

373 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
196 -> 373 [arrowhead = none ];
373 -> 198;

374 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
196 -> 374 [arrowhead = none ];
374 -> 199;

375 [label="EVAL with clause\napp(.(X318, X319), X320, .(X318, X321)) :- app(X319, X320, X321).\nand substitutionX318 -> T166,\nX319 -> T167,\nT138 -> .(T166, T167),\nT126 -> T168,\nT143 -> T169,\nX320 -> .(T168, T169),\nX321 -> X322,\nX246 -> .(T166, X322)", fontsize=14, style = filled, fillcolor = lightblue];
198 -> 375 [arrowhead = none ];
375 -> 204;

376 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
198 -> 376 [arrowhead = none ];
376 -> 205;

377 [label="EVAL with clause\napp([], X330, X330).\nand substitutionT138 -> [],\nT126 -> T178,\nT143 -> T179,\nX330 -> .(T178, T179),\nX246 -> .(T178, T179)", fontsize=14, style = filled, fillcolor = lightblue];
199 -> 377 [arrowhead = none ];
377 -> 210;

378 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
199 -> 378 [arrowhead = none ];
378 -> 211;

379 [label="INSTANCE with matching:\nT138 -> T167\nT126 -> T168\nT143 -> T169\nX246 -> X322", fontsize=14, style = filled, fillcolor = lightgrey];
204 -> 379 [arrowhead = none , style=dashed];
379 -> 189[style=dashed];

380 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
210 -> 380 [arrowhead = none ];
380 -> 212;

381 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
213 -> 381 [arrowhead = none ];
381 -> 216;

382 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
222 -> 382 [arrowhead = none ];
382 -> 225;

383 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
222 -> 383 [arrowhead = none ];
383 -> 226;

384 [label="EVAL with clause\napp(.(X351, X352), X353, .(X351, X354)) :- app(X352, X353, X354).\nand substitutionX351 -> T205,\nX352 -> T206,\nT110 -> .(T205, T206),\nT22 -> T207,\nT115 -> T208,\nX353 -> .(T207, T208),\nX354 -> T210,\nT9 -> .(T205, T210),\nT209 -> T210", fontsize=14, style = filled, fillcolor = lightblue];
225 -> 384 [arrowhead = none ];
384 -> 228;

385 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
225 -> 385 [arrowhead = none ];
385 -> 229;

386 [label="EVAL with clause\napp([], X361, X361).\nand substitutionT110 -> [],\nT22 -> T220,\nT115 -> T221,\nX361 -> .(T220, T221),\nT9 -> .(T220, T221)", fontsize=14, style = filled, fillcolor = lightblue];
226 -> 386 [arrowhead = none ];
386 -> 232;

387 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
226 -> 387 [arrowhead = none ];
387 -> 233;

388 [label="INSTANCE with matching:\nT110 -> T206\nT22 -> T207\nT115 -> T208\nT9 -> T210", fontsize=14, style = filled, fillcolor = lightgrey];
228 -> 388 [arrowhead = none , style=dashed];
388 -> 140[style=dashed];

389 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
232 -> 389 [arrowhead = none ];
389 -> 234;

390 [label="EVAL with clause\npart(X390, .(X391, X392), X393, .(X391, X394)) :- ','(le(X390, X391), part(X390, X392, X393, X394)).\nand substitutionT6 -> T234,\nX390 -> T234,\nX391 -> T235,\nX392 -> T236,\nT7 -> .(T235, T236),\nX7 -> X395,\nX393 -> X395,\nX394 -> X396,\nX8 -> .(T235, X396)", fontsize=14, style = filled, fillcolor = lightblue];
237 -> 390 [arrowhead = none ];
390 -> 246;

391 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
237 -> 391 [arrowhead = none ];
391 -> 247;

392 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
238 -> 392 [arrowhead = none ];
392 -> 267;

393 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
238 -> 393 [arrowhead = none ];
393 -> 268;

394 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
246 -> 394 [arrowhead = none ];
394 -> 250;

395 [label="SPLIT 2\nnew knowledge:\nT234 is ground\nT235 is ground", fontsize=14, style = filled, fillcolor = violet];
246 -> 395 [arrowhead = none ];
395 -> 251;

396 [label="INSTANCE with matching:\nT83 -> T234\nT84 -> T235", fontsize=14, style = filled, fillcolor = lightgrey];
250 -> 396 [arrowhead = none , style=dashed];
396 -> 102[style=dashed];

397 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
251 -> 397 [arrowhead = none ];
397 -> 257;

398 [label="SPLIT 2\nnew knowledge:\nT234 is ground\nT236 is ground\nT244 is ground\nT245 is ground\nreplacements:X395 -> T244,\nX396 -> T245", fontsize=14, style = filled, fillcolor = violet];
251 -> 398 [arrowhead = none ];
398 -> 258;

399 [label="INSTANCE with matching:\nT22 -> T234\nT24 -> T236\nX44 -> X395\nX45 -> X396", fontsize=14, style = filled, fillcolor = lightgrey];
257 -> 399 [arrowhead = none , style=dashed];
399 -> 57[style=dashed];

400 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
258 -> 400 [arrowhead = none ];
400 -> 259;

401 [label="SPLIT 2\nnew knowledge:\nT244 is ground\nT251 is ground\nreplacements:X9 -> T251", fontsize=14, style = filled, fillcolor = violet];
258 -> 401 [arrowhead = none ];
401 -> 262;

402 [label="INSTANCE with matching:\nT45 -> T244\nX10 -> X9", fontsize=14, style = filled, fillcolor = lightgrey];
259 -> 402 [arrowhead = none , style=dashed];
402 -> 139[style=dashed];

403 [label="INSTANCE with matching:\nT45 -> .(T235, T245)\nT110 -> T251\nT22 -> T234", fontsize=14, style = filled, fillcolor = lightgrey];
262 -> 403 [arrowhead = none , style=dashed];
403 -> 134[style=dashed];

404 [label="EVAL with clause\npart(X443, [], [], []).\nand substitutionT6 -> T260,\nX443 -> T260,\nT7 -> [],\nX7 -> [],\nX8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
267 -> 404 [arrowhead = none ];
404 -> 269;

405 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
267 -> 405 [arrowhead = none ];
405 -> 270;

406 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
268 -> 406 [arrowhead = none ];
406 -> 287;

407 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
269 -> 407 [arrowhead = none ];
407 -> 273;

408 [label="SPLIT 2\nnew knowledge:\nT261 is ground\nreplacements:X9 -> T261", fontsize=14, style = filled, fillcolor = violet];
269 -> 408 [arrowhead = none ];
408 -> 274;

409 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
273 -> 409 [arrowhead = none ];
409 -> 275;

410 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
274 -> 410 [arrowhead = none ];
410 -> 281;

411 [label="SPLIT 2\nnew knowledge:\nT262 is ground\nreplacements:X10 -> T262", fontsize=14, style = filled, fillcolor = violet];
274 -> 411 [arrowhead = none ];
411 -> 282;

412 [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];
275 -> 412 [arrowhead = none ];
412 -> 276;

413 [label="ONLY EVAL with clause\nqs([], []).\nand substitutionX9 -> []", fontsize=14, style = filled, fillcolor = lightblue];
276 -> 413 [arrowhead = none ];
413 -> 277;

414 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
277 -> 414 [arrowhead = none ];
414 -> 278;

415 [label="INSTANCE with matching:\nX9 -> X10", fontsize=14, style = filled, fillcolor = lightgrey];
281 -> 415 [arrowhead = none , style=dashed];
415 -> 273[style=dashed];

416 [label="INSTANCE with matching:\nT110 -> T261\nT22 -> T260\nT115 -> T262", fontsize=14, style = filled, fillcolor = lightgrey];
282 -> 416 [arrowhead = none , style=dashed];
416 -> 140[style=dashed];

417 [label="BACKTRACK\nfor clause: qs([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
287 -> 417 [arrowhead = none ];
417 -> 288;

418 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
289 -> 418 [arrowhead = none ];
418 -> 291;

}

(2) Obligation:

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

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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, X44, X45, X9, X10, T9))
QSA_IN_GA(.(T22, .(T23, T24)), T9) → PB_IN_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9)
PB_IN_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9) → U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
PB_IN_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9) → GTE_IN_GG(T22, T23)
GTE_IN_GG(s(T37), s(T38)) → U4_GG(T37, T38, gtE_in_gg(T37, T38))
GTE_IN_GG(s(T37), s(T38)) → GTE_IN_GG(T37, T38)
U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → PO_IN_GGAAGAAA(T22, T24, X44, X45, T23, X9, X10, T9)
PO_IN_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9) → U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
PO_IN_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9) → PARTF_IN_GGAA(T22, T24, T44, T45)
PARTF_IN_GGAA(T61, .(T62, T63), .(T62, X109), X110) → U5_GGAA(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
PARTF_IN_GGAA(T61, .(T62, T63), .(T62, X109), X110) → PG_IN_GGGAA(T61, T62, T63, X109, X110)
PG_IN_GGGAA(T61, T62, T63, X109, X110) → U17_GGGAA(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
PG_IN_GGGAA(T61, T62, T63, X109, X110) → GTE_IN_GG(T61, T62)
U17_GGGAA(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_GGGAA(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
U17_GGGAA(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → PARTF_IN_GGAA(T61, T63, X109, X110)
PARTF_IN_GGAA(T83, .(T84, T85), X155, .(T84, X156)) → U6_GGAA(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
PARTF_IN_GGAA(T83, .(T84, T85), X155, .(T84, X156)) → PH_IN_GGGAA(T83, T84, T85, X155, X156)
PH_IN_GGGAA(T83, T84, T85, X155, X156) → U19_GGGAA(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
PH_IN_GGGAA(T83, T84, T85, X155, X156) → LEI_IN_GG(T83, T84)
LEI_IN_GG(s(T98), s(T99)) → U7_GG(T98, T99, leI_in_gg(T98, T99))
LEI_IN_GG(s(T98), s(T99)) → LEI_IN_GG(T98, T99)
U19_GGGAA(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_GGGAA(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
U19_GGGAA(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → PARTF_IN_GGAA(T83, T85, X155, X156)
U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → PP_IN_GGAGAGA(T23, T44, X9, T45, X10, T22, T9)
PP_IN_GGAGAGA(T23, T44, T110, T45, X10, T22, T9) → U15_GGAGAGA(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
PP_IN_GGAGAGA(T23, T44, T110, T45, X10, T22, T9) → QSA_IN_GA(.(T23, T44), T110)
QSA_IN_GA(.(T234, .(T235, T236)), T9) → U2_GA(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
QSA_IN_GA(.(T234, .(T235, T236)), T9) → PC_IN_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9)
PC_IN_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9) → U29_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
PC_IN_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9) → LEI_IN_GG(T234, T235)
U29_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
U29_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → PT_IN_GGAAAGAA(T234, T236, X395, X396, X9, T235, X10, T9)
PT_IN_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9) → U31_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
PT_IN_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9) → PARTF_IN_GGAA(T234, T236, T244, T245)
U31_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
U31_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → PU_IN_GAGGAGA(T244, X9, T235, T245, X10, T234, T9)
PU_IN_GAGGAGA(T244, T251, T235, T245, X10, T234, T9) → U33_GAGGAGA(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
PU_IN_GAGGAGA(T244, T251, T235, T245, X10, T234, T9) → QSJ_IN_GA(T244, T251)
QSJ_IN_GA(.(T126, T127), X246) → U8_GA(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
QSJ_IN_GA(.(T126, T127), X246) → PK_IN_GGAAAAA(T126, T127, X242, X243, X244, X245, X246)
PK_IN_GGAAAAA(T126, T127, T131, T132, X244, X245, X246) → U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
PK_IN_GGAAAAA(T126, T127, T131, T132, X244, X245, X246) → PARTF_IN_GGAA(T126, T127, T131, T132)
U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → PR_IN_GAGAGA(T131, X244, T132, X245, T126, X246)
PR_IN_GAGAGA(T131, T138, T132, X245, T126, X246) → U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
PR_IN_GAGAGA(T131, T138, T132, X245, T126, X246) → QSJ_IN_GA(T131, T138)
U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_GAGAGA(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → PS_IN_GAGGA(T132, X245, T138, T126, X246)
PS_IN_GAGGA(T132, T143, T138, T126, X246) → U27_GAGGA(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
PS_IN_GAGGA(T132, T143, T138, T126, X246) → QSJ_IN_GA(T132, T143)
U27_GAGGA(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_GAGGA(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
U27_GAGGA(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → APPL_IN_GGGA(T138, T126, T143, X246)
APPL_IN_GGGA(.(T166, T167), T168, T169, .(T166, X322)) → U9_GGGA(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
APPL_IN_GGGA(.(T166, T167), T168, T169, .(T166, X322)) → APPL_IN_GGGA(T167, T168, T169, X322)
U33_GAGGAGA(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_GAGGAGA(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
U33_GAGGAGA(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → PQ_IN_GAGGA(.(T235, T245), X10, T251, T234, T9)
PQ_IN_GAGGA(T45, T115, T110, T22, T9) → U21_GAGGA(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
PQ_IN_GAGGA(T45, T115, T110, T22, T9) → QSJ_IN_GA(T45, T115)
U21_GAGGA(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_GAGGA(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
U21_GAGGA(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → APPM_IN_GGGA(T110, T22, T115, T9)
APPM_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → U10_GGGA(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
APPM_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → APPM_IN_GGGA(T206, T207, T208, T210)
QSA_IN_GA(.(T260, []), T9) → U3_GA(T260, T9, pD_in_aaga(X9, X10, T260, T9))
QSA_IN_GA(.(T260, []), T9) → PD_IN_AAGA(X9, X10, T260, T9)
PD_IN_AAGA(T261, X10, T260, T9) → U35_AAGA(T261, X10, T260, T9, qsN_in_a(T261))
PD_IN_AAGA(T261, X10, T260, T9) → QSN_IN_A(T261)
U35_AAGA(T261, X10, T260, T9, qsN_out_a(T261)) → U36_AAGA(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
U35_AAGA(T261, X10, T260, T9, qsN_out_a(T261)) → PV_IN_AGGA(X10, T261, T260, T9)
PV_IN_AGGA(T262, T261, T260, T9) → U37_AGGA(T262, T261, T260, T9, qsN_in_a(T262))
PV_IN_AGGA(T262, T261, T260, T9) → QSN_IN_A(T262)
U37_AGGA(T262, T261, T260, T9, qsN_out_a(T262)) → U38_AGGA(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U37_AGGA(T262, T261, T260, T9, qsN_out_a(T262)) → APPM_IN_GGGA(T261, T260, T262, T9)
U15_GGAGAGA(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_GGAGAGA(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U15_GGAGAGA(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → PQ_IN_GAGGA(T45, X10, T110, T22, T9)

The TRS R consists of the following rules:

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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)
U11_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_GGGAAAAA(x1, x2, x3, x9)
GTE_IN_GG(x1, x2)  =  GTE_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
U12_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_GGGAAAAA(x1, x2, x3, x9)
PO_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PO_IN_GGAAGAAA(x1, x2, x5)
U13_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_GGGAA(x1, x2, x3, x4, x5, x6)  =  U17_GGGAA(x1, x2, x3, x6)
U18_GGGAA(x1, x2, x3, x4, x5, x6)  =  U18_GGGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
PH_IN_GGGAA(x1, x2, x3, x4, x5)  =  PH_IN_GGGAA(x1, x2, x3)
U19_GGGAA(x1, x2, x3, x4, x5, x6)  =  U19_GGGAA(x1, x2, x3, x6)
LEI_IN_GG(x1, x2)  =  LEI_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U20_GGGAA(x1, x2, x3, x4, x5, x6)  =  U20_GGGAA(x1, x2, x3, x6)
U14_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_GGAAGAAA(x1, x2, x3, x4, x5, x9)
PP_IN_GGAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PP_IN_GGAGAGA(x1, x2, x4, x6)
U15_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GGAGAGA(x1, x2, x4, x6, x8)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
PC_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PC_IN_GGGAAAAA(x1, x2, x3)
U29_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_GGGAAAAA(x1, x2, x3, x9)
U30_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_GGGAAAAA(x1, x2, x3, x9)
PT_IN_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PT_IN_GGAAAGAA(x1, x2, x6)
U31_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_GGAAAGAA(x1, x2, x6, x9)
U32_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_GGAAAGAA(x1, x2, x3, x4, x6, x9)
PU_IN_GAGGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PU_IN_GAGGAGA(x1, x3, x4, x6)
U33_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_GAGGAGA(x1, x3, x4, x6, x8)
QSJ_IN_GA(x1, x2)  =  QSJ_IN_GA(x1)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
PK_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PK_IN_GGAAAAA(x1, x2)
U23_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_GGAAAAA(x1, x2, x8)
U24_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_GGAAAAA(x1, x2, x3, x4, x8)
PR_IN_GAGAGA(x1, x2, x3, x4, x5, x6)  =  PR_IN_GAGAGA(x1, x3, x5)
U25_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U25_GAGAGA(x1, x3, x5, x7)
U26_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U26_GAGAGA(x1, x2, x3, x5, x7)
PS_IN_GAGGA(x1, x2, x3, x4, x5)  =  PS_IN_GAGGA(x1, x3, x4)
U27_GAGGA(x1, x2, x3, x4, x5, x6)  =  U27_GAGGA(x1, x3, x4, x6)
U28_GAGGA(x1, x2, x3, x4, x5, x6)  =  U28_GAGGA(x1, x2, x3, x4, x6)
APPL_IN_GGGA(x1, x2, x3, x4)  =  APPL_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6)  =  U9_GGGA(x1, x2, x3, x4, x6)
U34_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_GAGGAGA(x1, x2, x3, x4, x6, x8)
PQ_IN_GAGGA(x1, x2, x3, x4, x5)  =  PQ_IN_GAGGA(x1, x3, x4)
U21_GAGGA(x1, x2, x3, x4, x5, x6)  =  U21_GAGGA(x1, x3, x4, x6)
U22_GAGGA(x1, x2, x3, x4, x5, x6)  =  U22_GAGGA(x1, x2, x3, x4, x6)
APPM_IN_GGGA(x1, x2, x3, x4)  =  APPM_IN_GGGA(x1, x2, x3)
U10_GGGA(x1, x2, x3, x4, x5, x6)  =  U10_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)
U35_AAGA(x1, x2, x3, x4, x5)  =  U35_AAGA(x3, x5)
QSN_IN_A(x1)  =  QSN_IN_A
U36_AAGA(x1, x2, x3, x4, x5)  =  U36_AAGA(x1, x3, x5)
PV_IN_AGGA(x1, x2, x3, x4)  =  PV_IN_AGGA(x2, x3)
U37_AGGA(x1, x2, x3, x4, x5)  =  U37_AGGA(x2, x3, x5)
U38_AGGA(x1, x2, x3, x4, x5)  =  U38_AGGA(x1, x2, x3, x5)
U16_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_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, X44, X45, X9, X10, T9))
QSA_IN_GA(.(T22, .(T23, T24)), T9) → PB_IN_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9)
PB_IN_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9) → U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
PB_IN_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9) → GTE_IN_GG(T22, T23)
GTE_IN_GG(s(T37), s(T38)) → U4_GG(T37, T38, gtE_in_gg(T37, T38))
GTE_IN_GG(s(T37), s(T38)) → GTE_IN_GG(T37, T38)
U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → PO_IN_GGAAGAAA(T22, T24, X44, X45, T23, X9, X10, T9)
PO_IN_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9) → U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
PO_IN_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9) → PARTF_IN_GGAA(T22, T24, T44, T45)
PARTF_IN_GGAA(T61, .(T62, T63), .(T62, X109), X110) → U5_GGAA(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
PARTF_IN_GGAA(T61, .(T62, T63), .(T62, X109), X110) → PG_IN_GGGAA(T61, T62, T63, X109, X110)
PG_IN_GGGAA(T61, T62, T63, X109, X110) → U17_GGGAA(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
PG_IN_GGGAA(T61, T62, T63, X109, X110) → GTE_IN_GG(T61, T62)
U17_GGGAA(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_GGGAA(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
U17_GGGAA(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → PARTF_IN_GGAA(T61, T63, X109, X110)
PARTF_IN_GGAA(T83, .(T84, T85), X155, .(T84, X156)) → U6_GGAA(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
PARTF_IN_GGAA(T83, .(T84, T85), X155, .(T84, X156)) → PH_IN_GGGAA(T83, T84, T85, X155, X156)
PH_IN_GGGAA(T83, T84, T85, X155, X156) → U19_GGGAA(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
PH_IN_GGGAA(T83, T84, T85, X155, X156) → LEI_IN_GG(T83, T84)
LEI_IN_GG(s(T98), s(T99)) → U7_GG(T98, T99, leI_in_gg(T98, T99))
LEI_IN_GG(s(T98), s(T99)) → LEI_IN_GG(T98, T99)
U19_GGGAA(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_GGGAA(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
U19_GGGAA(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → PARTF_IN_GGAA(T83, T85, X155, X156)
U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → PP_IN_GGAGAGA(T23, T44, X9, T45, X10, T22, T9)
PP_IN_GGAGAGA(T23, T44, T110, T45, X10, T22, T9) → U15_GGAGAGA(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
PP_IN_GGAGAGA(T23, T44, T110, T45, X10, T22, T9) → QSA_IN_GA(.(T23, T44), T110)
QSA_IN_GA(.(T234, .(T235, T236)), T9) → U2_GA(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
QSA_IN_GA(.(T234, .(T235, T236)), T9) → PC_IN_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9)
PC_IN_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9) → U29_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
PC_IN_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9) → LEI_IN_GG(T234, T235)
U29_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
U29_GGGAAAAA(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → PT_IN_GGAAAGAA(T234, T236, X395, X396, X9, T235, X10, T9)
PT_IN_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9) → U31_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
PT_IN_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9) → PARTF_IN_GGAA(T234, T236, T244, T245)
U31_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
U31_GGAAAGAA(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → PU_IN_GAGGAGA(T244, X9, T235, T245, X10, T234, T9)
PU_IN_GAGGAGA(T244, T251, T235, T245, X10, T234, T9) → U33_GAGGAGA(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
PU_IN_GAGGAGA(T244, T251, T235, T245, X10, T234, T9) → QSJ_IN_GA(T244, T251)
QSJ_IN_GA(.(T126, T127), X246) → U8_GA(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
QSJ_IN_GA(.(T126, T127), X246) → PK_IN_GGAAAAA(T126, T127, X242, X243, X244, X245, X246)
PK_IN_GGAAAAA(T126, T127, T131, T132, X244, X245, X246) → U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
PK_IN_GGAAAAA(T126, T127, T131, T132, X244, X245, X246) → PARTF_IN_GGAA(T126, T127, T131, T132)
U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → PR_IN_GAGAGA(T131, X244, T132, X245, T126, X246)
PR_IN_GAGAGA(T131, T138, T132, X245, T126, X246) → U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
PR_IN_GAGAGA(T131, T138, T132, X245, T126, X246) → QSJ_IN_GA(T131, T138)
U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_GAGAGA(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → PS_IN_GAGGA(T132, X245, T138, T126, X246)
PS_IN_GAGGA(T132, T143, T138, T126, X246) → U27_GAGGA(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
PS_IN_GAGGA(T132, T143, T138, T126, X246) → QSJ_IN_GA(T132, T143)
U27_GAGGA(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_GAGGA(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
U27_GAGGA(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → APPL_IN_GGGA(T138, T126, T143, X246)
APPL_IN_GGGA(.(T166, T167), T168, T169, .(T166, X322)) → U9_GGGA(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
APPL_IN_GGGA(.(T166, T167), T168, T169, .(T166, X322)) → APPL_IN_GGGA(T167, T168, T169, X322)
U33_GAGGAGA(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_GAGGAGA(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
U33_GAGGAGA(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → PQ_IN_GAGGA(.(T235, T245), X10, T251, T234, T9)
PQ_IN_GAGGA(T45, T115, T110, T22, T9) → U21_GAGGA(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
PQ_IN_GAGGA(T45, T115, T110, T22, T9) → QSJ_IN_GA(T45, T115)
U21_GAGGA(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_GAGGA(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
U21_GAGGA(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → APPM_IN_GGGA(T110, T22, T115, T9)
APPM_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → U10_GGGA(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
APPM_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → APPM_IN_GGGA(T206, T207, T208, T210)
QSA_IN_GA(.(T260, []), T9) → U3_GA(T260, T9, pD_in_aaga(X9, X10, T260, T9))
QSA_IN_GA(.(T260, []), T9) → PD_IN_AAGA(X9, X10, T260, T9)
PD_IN_AAGA(T261, X10, T260, T9) → U35_AAGA(T261, X10, T260, T9, qsN_in_a(T261))
PD_IN_AAGA(T261, X10, T260, T9) → QSN_IN_A(T261)
U35_AAGA(T261, X10, T260, T9, qsN_out_a(T261)) → U36_AAGA(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
U35_AAGA(T261, X10, T260, T9, qsN_out_a(T261)) → PV_IN_AGGA(X10, T261, T260, T9)
PV_IN_AGGA(T262, T261, T260, T9) → U37_AGGA(T262, T261, T260, T9, qsN_in_a(T262))
PV_IN_AGGA(T262, T261, T260, T9) → QSN_IN_A(T262)
U37_AGGA(T262, T261, T260, T9, qsN_out_a(T262)) → U38_AGGA(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U37_AGGA(T262, T261, T260, T9, qsN_out_a(T262)) → APPM_IN_GGGA(T261, T260, T262, T9)
U15_GGAGAGA(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_GGAGAGA(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U15_GGAGAGA(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → PQ_IN_GAGGA(T45, X10, T110, T22, T9)

The TRS R consists of the following rules:

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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)
U11_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_GGGAAAAA(x1, x2, x3, x9)
GTE_IN_GG(x1, x2)  =  GTE_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
U12_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_GGGAAAAA(x1, x2, x3, x9)
PO_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PO_IN_GGAAGAAA(x1, x2, x5)
U13_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_GGGAA(x1, x2, x3, x4, x5, x6)  =  U17_GGGAA(x1, x2, x3, x6)
U18_GGGAA(x1, x2, x3, x4, x5, x6)  =  U18_GGGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
PH_IN_GGGAA(x1, x2, x3, x4, x5)  =  PH_IN_GGGAA(x1, x2, x3)
U19_GGGAA(x1, x2, x3, x4, x5, x6)  =  U19_GGGAA(x1, x2, x3, x6)
LEI_IN_GG(x1, x2)  =  LEI_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U20_GGGAA(x1, x2, x3, x4, x5, x6)  =  U20_GGGAA(x1, x2, x3, x6)
U14_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_GGAAGAAA(x1, x2, x3, x4, x5, x9)
PP_IN_GGAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PP_IN_GGAGAGA(x1, x2, x4, x6)
U15_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GGAGAGA(x1, x2, x4, x6, x8)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
PC_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PC_IN_GGGAAAAA(x1, x2, x3)
U29_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_GGGAAAAA(x1, x2, x3, x9)
U30_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_GGGAAAAA(x1, x2, x3, x9)
PT_IN_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PT_IN_GGAAAGAA(x1, x2, x6)
U31_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_GGAAAGAA(x1, x2, x6, x9)
U32_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_GGAAAGAA(x1, x2, x3, x4, x6, x9)
PU_IN_GAGGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PU_IN_GAGGAGA(x1, x3, x4, x6)
U33_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_GAGGAGA(x1, x3, x4, x6, x8)
QSJ_IN_GA(x1, x2)  =  QSJ_IN_GA(x1)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
PK_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PK_IN_GGAAAAA(x1, x2)
U23_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_GGAAAAA(x1, x2, x8)
U24_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_GGAAAAA(x1, x2, x3, x4, x8)
PR_IN_GAGAGA(x1, x2, x3, x4, x5, x6)  =  PR_IN_GAGAGA(x1, x3, x5)
U25_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U25_GAGAGA(x1, x3, x5, x7)
U26_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U26_GAGAGA(x1, x2, x3, x5, x7)
PS_IN_GAGGA(x1, x2, x3, x4, x5)  =  PS_IN_GAGGA(x1, x3, x4)
U27_GAGGA(x1, x2, x3, x4, x5, x6)  =  U27_GAGGA(x1, x3, x4, x6)
U28_GAGGA(x1, x2, x3, x4, x5, x6)  =  U28_GAGGA(x1, x2, x3, x4, x6)
APPL_IN_GGGA(x1, x2, x3, x4)  =  APPL_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6)  =  U9_GGGA(x1, x2, x3, x4, x6)
U34_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_GAGGAGA(x1, x2, x3, x4, x6, x8)
PQ_IN_GAGGA(x1, x2, x3, x4, x5)  =  PQ_IN_GAGGA(x1, x3, x4)
U21_GAGGA(x1, x2, x3, x4, x5, x6)  =  U21_GAGGA(x1, x3, x4, x6)
U22_GAGGA(x1, x2, x3, x4, x5, x6)  =  U22_GAGGA(x1, x2, x3, x4, x6)
APPM_IN_GGGA(x1, x2, x3, x4)  =  APPM_IN_GGGA(x1, x2, x3)
U10_GGGA(x1, x2, x3, x4, x5, x6)  =  U10_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)
U35_AAGA(x1, x2, x3, x4, x5)  =  U35_AAGA(x3, x5)
QSN_IN_A(x1)  =  QSN_IN_A
U36_AAGA(x1, x2, x3, x4, x5)  =  U36_AAGA(x1, x3, x5)
PV_IN_AGGA(x1, x2, x3, x4)  =  PV_IN_AGGA(x2, x3)
U37_AGGA(x1, x2, x3, x4, x5)  =  U37_AGGA(x2, x3, x5)
U38_AGGA(x1, x2, x3, x4, x5)  =  U38_AGGA(x1, x2, x3, x5)
U16_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_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 7 SCCs with 53 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPM_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → APPM_IN_GGGA(T206, T207, T208, T210)

The TRS R consists of the following rules:

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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)
APPM_IN_GGGA(x1, x2, x3, x4)  =  APPM_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:

APPM_IN_GGGA(.(T205, T206), T207, T208, .(T205, T210)) → APPM_IN_GGGA(T206, T207, T208, T210)

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

APPM_IN_GGGA(.(T205, T206), T207, T208) → APPM_IN_GGGA(T206, T207, T208)

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:

• APPM_IN_GGGA(.(T205, T206), T207, T208) → APPM_IN_GGGA(T206, T207, T208)
  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:

APPL_IN_GGGA(.(T166, T167), T168, T169, .(T166, X322)) → APPL_IN_GGGA(T167, T168, T169, X322)

The TRS R consists of the following rules:

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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)
APPL_IN_GGGA(x1, x2, x3, x4)  =  APPL_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:

APPL_IN_GGGA(.(T166, T167), T168, T169, .(T166, X322)) → APPL_IN_GGGA(T167, T168, T169, X322)

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

APPL_IN_GGGA(.(T166, T167), T168, T169) → APPL_IN_GGGA(T167, T168, T169)

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:

• APPL_IN_GGGA(.(T166, T167), T168, T169) → APPL_IN_GGGA(T167, T168, T169)
  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:

LEI_IN_GG(s(T98), s(T99)) → LEI_IN_GG(T98, T99)

The TRS R consists of the following rules:

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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)
LEI_IN_GG(x1, x2)  =  LEI_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:

LEI_IN_GG(s(T98), s(T99)) → LEI_IN_GG(T98, T99)

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:

LEI_IN_GG(s(T98), s(T99)) → LEI_IN_GG(T98, T99)

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:

• LEI_IN_GG(s(T98), s(T99)) → LEI_IN_GG(T98, T99)
  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:

GTE_IN_GG(s(T37), s(T38)) → GTE_IN_GG(T37, T38)

The TRS R consists of the following rules:

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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)
GTE_IN_GG(x1, x2)  =  GTE_IN_GG(x1, x2)

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:

GTE_IN_GG(s(T37), s(T38)) → GTE_IN_GG(T37, T38)

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

(31) PiDPToQDPProof (EQUIVALENT 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:

GTE_IN_GG(s(T37), s(T38)) → GTE_IN_GG(T37, T38)

R is empty.
Q is empty.
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:

• GTE_IN_GG(s(T37), s(T38)) → GTE_IN_GG(T37, T38)
  The graph contains the following edges 1 > 1, 2 > 2

(34) YES

(35) Obligation:

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

PG_IN_GGGAA(T61, T62, T63, X109, X110) → U17_GGGAA(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_GGGAA(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → PARTF_IN_GGAA(T61, T63, X109, X110)
PARTF_IN_GGAA(T61, .(T62, T63), .(T62, X109), X110) → PG_IN_GGGAA(T61, T62, T63, X109, X110)
PARTF_IN_GGAA(T83, .(T84, T85), X155, .(T84, X156)) → PH_IN_GGGAA(T83, T84, T85, X155, X156)
PH_IN_GGGAA(T83, T84, T85, X155, X156) → U19_GGGAA(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
U19_GGGAA(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → PARTF_IN_GGAA(T83, T85, X155, X156)

The TRS R consists of the following rules:

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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)
U17_GGGAA(x1, x2, x3, x4, x5, x6)  =  U17_GGGAA(x1, x2, x3, x6)
PH_IN_GGGAA(x1, x2, x3, x4, x5)  =  PH_IN_GGGAA(x1, x2, x3)
U19_GGGAA(x1, x2, x3, x4, x5, x6)  =  U19_GGGAA(x1, x2, x3, x6)

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:

PG_IN_GGGAA(T61, T62, T63, X109, X110) → U17_GGGAA(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_GGGAA(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → PARTF_IN_GGAA(T61, T63, X109, X110)
PARTF_IN_GGAA(T61, .(T62, T63), .(T62, X109), X110) → PG_IN_GGGAA(T61, T62, T63, X109, X110)
PARTF_IN_GGAA(T83, .(T84, T85), X155, .(T84, X156)) → PH_IN_GGGAA(T83, T84, T85, X155, X156)
PH_IN_GGGAA(T83, T84, T85, X155, X156) → U19_GGGAA(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
U19_GGGAA(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → PARTF_IN_GGAA(T83, T85, X155, X156)

The TRS R consists of the following rules:

gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
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)
U17_GGGAA(x1, x2, x3, x4, x5, x6)  =  U17_GGGAA(x1, x2, x3, x6)
PH_IN_GGGAA(x1, x2, x3, x4, x5)  =  PH_IN_GGGAA(x1, x2, x3)
U19_GGGAA(x1, x2, x3, x4, x5, x6)  =  U19_GGGAA(x1, x2, x3, x6)

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:

PG_IN_GGGAA(T61, T62, T63) → U17_GGGAA(T61, T62, T63, gtE_in_gg(T61, T62))
U17_GGGAA(T61, T62, T63, gtE_out_gg(T61, T62)) → PARTF_IN_GGAA(T61, T63)
PARTF_IN_GGAA(T61, .(T62, T63)) → PG_IN_GGGAA(T61, T62, T63)
PARTF_IN_GGAA(T83, .(T84, T85)) → PH_IN_GGGAA(T83, T84, T85)
PH_IN_GGGAA(T83, T84, T85) → U19_GGGAA(T83, T84, T85, leI_in_gg(T83, T84))
U19_GGGAA(T83, T84, T85, leI_out_gg(T83, T84)) → PARTF_IN_GGAA(T83, T85)

The TRS R consists of the following rules:

gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))

The set Q consists of the following terms:

gtE_in_gg(x0, x1)
leI_in_gg(x0, x1)
U4_gg(x0, x1, x2)
U7_gg(x0, x1, x2)

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

(40) 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:

• U17_GGGAA(T61, T62, T63, gtE_out_gg(T61, T62)) → PARTF_IN_GGAA(T61, T63)
  The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

• PARTF_IN_GGAA(T61, .(T62, T63)) → PG_IN_GGGAA(T61, T62, T63)
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

• PARTF_IN_GGAA(T83, .(T84, T85)) → PH_IN_GGGAA(T83, T84, T85)
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

• PG_IN_GGGAA(T61, T62, T63) → U17_GGGAA(T61, T62, T63, gtE_in_gg(T61, T62))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

• U19_GGGAA(T83, T84, T85, leI_out_gg(T83, T84)) → PARTF_IN_GGAA(T83, T85)
  The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

• PH_IN_GGGAA(T83, T84, T85) → U19_GGGAA(T83, T84, T85, leI_in_gg(T83, T84))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

(41) YES

(42) Obligation:

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

PK_IN_GGAAAAA(T126, T127, T131, T132, X244, X245, X246) → U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → PR_IN_GAGAGA(T131, X244, T132, X245, T126, X246)
PR_IN_GAGAGA(T131, T138, T132, X245, T126, X246) → U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → PS_IN_GAGGA(T132, X245, T138, T126, X246)
PS_IN_GAGGA(T132, T143, T138, T126, X246) → QSJ_IN_GA(T132, T143)
QSJ_IN_GA(.(T126, T127), X246) → PK_IN_GGAAAAA(T126, T127, X242, X243, X244, X245, X246)
PR_IN_GAGAGA(T131, T138, T132, X245, T126, X246) → QSJ_IN_GA(T131, T138)

The TRS R consists of the following rules:

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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)
QSJ_IN_GA(x1, x2)  =  QSJ_IN_GA(x1)
PK_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PK_IN_GGAAAAA(x1, x2)
U23_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_GGAAAAA(x1, x2, x8)
PR_IN_GAGAGA(x1, x2, x3, x4, x5, x6)  =  PR_IN_GAGAGA(x1, x3, x5)
U25_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U25_GAGAGA(x1, x3, x5, x7)
PS_IN_GAGGA(x1, x2, x3, x4, x5)  =  PS_IN_GAGGA(x1, x3, x4)

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

(43) UsableRulesProof (EQUIVALENT transformation)

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

(44) Obligation:

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

PK_IN_GGAAAAA(T126, T127, T131, T132, X244, X245, X246) → U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_GGAAAAA(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → PR_IN_GAGAGA(T131, X244, T132, X245, T126, X246)
PR_IN_GAGAGA(T131, T138, T132, X245, T126, X246) → U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
U25_GAGAGA(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → PS_IN_GAGGA(T132, X245, T138, T126, X246)
PS_IN_GAGGA(T132, T143, T138, T126, X246) → QSJ_IN_GA(T132, T143)
QSJ_IN_GA(.(T126, T127), X246) → PK_IN_GGAAAAA(T126, T127, X242, X243, X244, X245, X246)
PR_IN_GAGAGA(T131, T138, T132, X245, T126, X246) → QSJ_IN_GA(T131, T138)

The TRS R consists of the following rules:

partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
QSJ_IN_GA(x1, x2)  =  QSJ_IN_GA(x1)
PK_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PK_IN_GGAAAAA(x1, x2)
U23_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_GGAAAAA(x1, x2, x8)
PR_IN_GAGAGA(x1, x2, x3, x4, x5, x6)  =  PR_IN_GAGAGA(x1, x3, x5)
U25_GAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  U25_GAGAGA(x1, x3, x5, x7)
PS_IN_GAGGA(x1, x2, x3, x4, x5)  =  PS_IN_GAGGA(x1, x3, x4)

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

(45) PiDPToQDPProof (SOUND transformation)

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

(46) Obligation:

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

PK_IN_GGAAAAA(T126, T127) → U23_GGAAAAA(T126, T127, partF_in_ggaa(T126, T127))
U23_GGAAAAA(T126, T127, partF_out_ggaa(T126, T127, T131, T132)) → PR_IN_GAGAGA(T131, T132, T126)
PR_IN_GAGAGA(T131, T132, T126) → U25_GAGAGA(T131, T132, T126, qsJ_in_ga(T131))
U25_GAGAGA(T131, T132, T126, qsJ_out_ga(T131, T138)) → PS_IN_GAGGA(T132, T138, T126)
PS_IN_GAGGA(T132, T138, T126) → QSJ_IN_GA(T132)
QSJ_IN_GA(.(T126, T127)) → PK_IN_GGAAAAA(T126, T127)
PR_IN_GAGAGA(T131, T132, T126) → QSJ_IN_GA(T131)

The TRS R consists of the following rules:

partF_in_ggaa(T61, .(T62, T63)) → U5_ggaa(T61, T62, T63, pG_in_gggaa(T61, T62, T63))
partF_in_ggaa(T83, .(T84, T85)) → U6_ggaa(T83, T84, T85, pH_in_gggaa(T83, T84, T85))
partF_in_ggaa(T107, []) → partF_out_ggaa(T107, [], [], [])
qsJ_in_ga(.(T126, T127)) → U8_ga(T126, T127, pK_in_ggaaaaa(T126, T127))
qsJ_in_ga([]) → qsJ_out_ga([], [])
U5_ggaa(T61, T62, T63, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U6_ggaa(T83, T84, T85, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U8_ga(T126, T127, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
pG_in_gggaa(T61, T62, T63) → U17_gggaa(T61, T62, T63, gtE_in_gg(T61, T62))
pH_in_gggaa(T83, T84, T85) → U19_gggaa(T83, T84, T85, leI_in_gg(T83, T84))
pK_in_ggaaaaa(T126, T127) → U23_ggaaaaa(T126, T127, partF_in_ggaa(T126, T127))
U17_gggaa(T61, T62, T63, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, partF_in_ggaa(T61, T63))
U19_gggaa(T83, T84, T85, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, partF_in_ggaa(T83, T85))
U23_ggaaaaa(T126, T127, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, pR_in_gagaga(T131, T132, T126))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U18_gggaa(T61, T62, T63, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U20_gggaa(T83, T84, T85, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U24_ggaaaaa(T126, T127, T131, T132, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
pR_in_gagaga(T131, T132, T126) → U25_gagaga(T131, T132, T126, qsJ_in_ga(T131))
U25_gagaga(T131, T132, T126, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, T126, pS_in_gagga(T132, T138, T126))
U26_gagaga(T131, T138, T132, T126, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
pS_in_gagga(T132, T138, T126) → U27_gagga(T132, T138, T126, qsJ_in_ga(T132))
U27_gagga(T132, T138, T126, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, appL_in_ggga(T138, T126, T143))
U28_gagga(T132, T143, T138, T126, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
appL_in_ggga(.(T166, T167), T168, T169) → U9_ggga(T166, T167, T168, T169, appL_in_ggga(T167, T168, T169))
appL_in_ggga([], T178, T179) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))

The set Q consists of the following terms:

partF_in_ggaa(x0, x1)
qsJ_in_ga(x0)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3)
U8_ga(x0, x1, x2)
pG_in_gggaa(x0, x1, x2)
pH_in_gggaa(x0, x1, x2)
pK_in_ggaaaaa(x0, x1)
U17_gggaa(x0, x1, x2, x3)
U19_gggaa(x0, x1, x2, x3)
U23_ggaaaaa(x0, x1, x2)
gtE_in_gg(x0, x1)
U18_gggaa(x0, x1, x2, x3)
leI_in_gg(x0, x1)
U20_gggaa(x0, x1, x2, x3)
U24_ggaaaaa(x0, x1, x2, x3, x4)
U4_gg(x0, x1, x2)
U7_gg(x0, x1, x2)
pR_in_gagaga(x0, x1, x2)
U25_gagaga(x0, x1, x2, x3)
U26_gagaga(x0, x1, x2, x3, x4)
pS_in_gagga(x0, x1, x2)
U27_gagga(x0, x1, x2, x3)
U28_gagga(x0, x1, x2, x3, x4)
appL_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(47) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PK_IN_GGAAAAA(T126, T127) → U23_GGAAAAA(T126, T127, partF_in_ggaa(T126, T127))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(0) = 0   
POL(PK_IN_GGAAAAA(x₁, x₂)) = 1 + x₂   
POL(PR_IN_GAGAGA(x₁, x₂, x₃)) = x₁ + x₂   
POL(PS_IN_GAGGA(x₁, x₂, x₃)) = x₁   
POL(QSJ_IN_GA(x₁)) = x₁   
POL(U17_gggaa(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U18_gggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U19_gggaa(x₁, x₂, x₃, x₄)) = x₃   
POL(U20_gggaa(x₁, x₂, x₃, x₄)) = x₄   
POL(U23_GGAAAAA(x₁, x₂, x₃)) = x₃   
POL(U23_ggaaaaa(x₁, x₂, x₃)) = 0   
POL(U24_ggaaaaa(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U25_GAGAGA(x₁, x₂, x₃, x₄)) = x₂   
POL(U25_gagaga(x₁, x₂, x₃, x₄)) = 0   
POL(U26_gagaga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U27_gagga(x₁, x₂, x₃, x₄)) = 0   
POL(U28_gagga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U4_gg(x₁, x₂, x₃)) = 0   
POL(U5_ggaa(x₁, x₂, x₃, x₄)) = x₄   
POL(U6_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U7_gg(x₁, x₂, x₃)) = 0   
POL(U8_ga(x₁, x₂, x₃)) = 0   
POL(U9_ggga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL([]) = 0   
POL(appL_in_ggga(x₁, x₂, x₃)) = 0   
POL(appL_out_ggga(x₁, x₂, x₃, x₄)) = 0   
POL(gtE_in_gg(x₁, x₂)) = 0   
POL(gtE_out_gg(x₁, x₂)) = 0   
POL(leI_in_gg(x₁, x₂)) = 0   
POL(leI_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 + x₅   
POL(pH_in_gggaa(x₁, x₂, x₃)) = x₃   
POL(pH_out_gggaa(x₁, x₂, x₃, x₄, x₅)) = x₄ + x₅   
POL(pK_in_ggaaaaa(x₁, x₂)) = 0   
POL(pK_out_ggaaaaa(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 0   
POL(pR_in_gagaga(x₁, x₂, x₃)) = 0   
POL(pR_out_gagaga(x₁, x₂, x₃, x₄, x₅, x₆)) = 0   
POL(pS_in_gagga(x₁, x₂, x₃)) = 0   
POL(pS_out_gagga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(partF_in_ggaa(x₁, x₂)) = x₂   
POL(partF_out_ggaa(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(qsJ_in_ga(x₁)) = 0   
POL(qsJ_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(T61, .(T62, T63)) → U5_ggaa(T61, T62, T63, pG_in_gggaa(T61, T62, T63))
partF_in_ggaa(T83, .(T84, T85)) → U6_ggaa(T83, T84, T85, pH_in_gggaa(T83, T84, T85))
partF_in_ggaa(T107, []) → partF_out_ggaa(T107, [], [], [])
pG_in_gggaa(T61, T62, T63) → U17_gggaa(T61, T62, T63, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, partF_in_ggaa(T61, T63))
U18_gggaa(T61, T62, T63, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U6_ggaa(T83, T84, T85, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
pH_in_gggaa(T83, T84, T85) → U19_gggaa(T83, T84, T85, leI_in_gg(T83, T84))
U19_gggaa(T83, T84, T85, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, partF_in_ggaa(T83, T85))
U20_gggaa(T83, T84, T85, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)

(48) Obligation:

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

U23_GGAAAAA(T126, T127, partF_out_ggaa(T126, T127, T131, T132)) → PR_IN_GAGAGA(T131, T132, T126)
PR_IN_GAGAGA(T131, T132, T126) → U25_GAGAGA(T131, T132, T126, qsJ_in_ga(T131))
U25_GAGAGA(T131, T132, T126, qsJ_out_ga(T131, T138)) → PS_IN_GAGGA(T132, T138, T126)
PS_IN_GAGGA(T132, T138, T126) → QSJ_IN_GA(T132)
QSJ_IN_GA(.(T126, T127)) → PK_IN_GGAAAAA(T126, T127)
PR_IN_GAGAGA(T131, T132, T126) → QSJ_IN_GA(T131)

The TRS R consists of the following rules:

partF_in_ggaa(T61, .(T62, T63)) → U5_ggaa(T61, T62, T63, pG_in_gggaa(T61, T62, T63))
partF_in_ggaa(T83, .(T84, T85)) → U6_ggaa(T83, T84, T85, pH_in_gggaa(T83, T84, T85))
partF_in_ggaa(T107, []) → partF_out_ggaa(T107, [], [], [])
qsJ_in_ga(.(T126, T127)) → U8_ga(T126, T127, pK_in_ggaaaaa(T126, T127))
qsJ_in_ga([]) → qsJ_out_ga([], [])
U5_ggaa(T61, T62, T63, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U6_ggaa(T83, T84, T85, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U8_ga(T126, T127, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
pG_in_gggaa(T61, T62, T63) → U17_gggaa(T61, T62, T63, gtE_in_gg(T61, T62))
pH_in_gggaa(T83, T84, T85) → U19_gggaa(T83, T84, T85, leI_in_gg(T83, T84))
pK_in_ggaaaaa(T126, T127) → U23_ggaaaaa(T126, T127, partF_in_ggaa(T126, T127))
U17_gggaa(T61, T62, T63, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, partF_in_ggaa(T61, T63))
U19_gggaa(T83, T84, T85, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, partF_in_ggaa(T83, T85))
U23_ggaaaaa(T126, T127, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, pR_in_gagaga(T131, T132, T126))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U18_gggaa(T61, T62, T63, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U20_gggaa(T83, T84, T85, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U24_ggaaaaa(T126, T127, T131, T132, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
pR_in_gagaga(T131, T132, T126) → U25_gagaga(T131, T132, T126, qsJ_in_ga(T131))
U25_gagaga(T131, T132, T126, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, T126, pS_in_gagga(T132, T138, T126))
U26_gagaga(T131, T138, T132, T126, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
pS_in_gagga(T132, T138, T126) → U27_gagga(T132, T138, T126, qsJ_in_ga(T132))
U27_gagga(T132, T138, T126, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, appL_in_ggga(T138, T126, T143))
U28_gagga(T132, T143, T138, T126, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
appL_in_ggga(.(T166, T167), T168, T169) → U9_ggga(T166, T167, T168, T169, appL_in_ggga(T167, T168, T169))
appL_in_ggga([], T178, T179) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))

The set Q consists of the following terms:

partF_in_ggaa(x0, x1)
qsJ_in_ga(x0)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3)
U8_ga(x0, x1, x2)
pG_in_gggaa(x0, x1, x2)
pH_in_gggaa(x0, x1, x2)
pK_in_ggaaaaa(x0, x1)
U17_gggaa(x0, x1, x2, x3)
U19_gggaa(x0, x1, x2, x3)
U23_ggaaaaa(x0, x1, x2)
gtE_in_gg(x0, x1)
U18_gggaa(x0, x1, x2, x3)
leI_in_gg(x0, x1)
U20_gggaa(x0, x1, x2, x3)
U24_ggaaaaa(x0, x1, x2, x3, x4)
U4_gg(x0, x1, x2)
U7_gg(x0, x1, x2)
pR_in_gagaga(x0, x1, x2)
U25_gagaga(x0, x1, x2, x3)
U26_gagaga(x0, x1, x2, x3, x4)
pS_in_gagga(x0, x1, x2)
U27_gagga(x0, x1, x2, x3)
U28_gagga(x0, x1, x2, x3, x4)
appL_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(49) DependencyGraphProof (EQUIVALENT transformation)

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

(50) TRUE

(51) 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, X44, X45, X9, X10, T9)
PB_IN_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9) → U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → PO_IN_GGAAGAAA(T22, T24, X44, X45, T23, X9, X10, T9)
PO_IN_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9) → U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → PP_IN_GGAGAGA(T23, T44, X9, T45, X10, T22, T9)
PP_IN_GGAGAGA(T23, T44, T110, T45, X10, T22, T9) → QSA_IN_GA(.(T23, T44), T110)

The TRS R consists of the following rules:

qsA_in_ga(.(T22, .(T23, T24)), T9) → U1_ga(T22, T23, T24, T9, pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9))
pB_in_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9) → U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U11_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_in_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9))
pO_in_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9) → U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U13_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_in_ggagaga(T23, T44, X9, T45, X10, T22, T9))
pP_in_ggagaga(T23, T44, T110, T45, X10, T22, T9) → U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_in_ga(.(T23, T44), T110))
qsA_in_ga(.(T234, .(T235, T236)), T9) → U2_ga(T234, T235, T236, T9, pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9))
pC_in_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9) → U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_in_gg(T234, T235))
U29_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, leI_out_gg(T234, T235)) → U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_in_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9))
pT_in_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9) → U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_in_ggaa(T234, T236, T244, T245))
U31_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, partF_out_ggaa(T234, T236, T244, T245)) → U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_in_gaggaga(T244, X9, T235, T245, X10, T234, T9))
pU_in_gaggaga(T244, T251, T235, T245, X10, T234, T9) → U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_in_ga(T244, T251))
qsJ_in_ga(.(T126, T127), X246) → U8_ga(T126, T127, X246, pK_in_ggaaaaa(T126, T127, X242, X243, X244, X245, X246))
pK_in_ggaaaaa(T126, T127, T131, T132, X244, X245, X246) → U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_in_ggaa(T126, T127, T131, T132))
U23_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, partF_out_ggaa(T126, T127, T131, T132)) → U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_in_gagaga(T131, X244, T132, X245, T126, X246))
pR_in_gagaga(T131, T138, T132, X245, T126, X246) → U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_in_ga(T131, T138))
qsJ_in_ga([], []) → qsJ_out_ga([], [])
U25_gagaga(T131, T138, T132, X245, T126, X246, qsJ_out_ga(T131, T138)) → U26_gagaga(T131, T138, T132, X245, T126, X246, pS_in_gagga(T132, X245, T138, T126, X246))
pS_in_gagga(T132, T143, T138, T126, X246) → U27_gagga(T132, T143, T138, T126, X246, qsJ_in_ga(T132, T143))
U27_gagga(T132, T143, T138, T126, X246, qsJ_out_ga(T132, T143)) → U28_gagga(T132, T143, T138, T126, X246, appL_in_ggga(T138, T126, T143, X246))
appL_in_ggga(.(T166, T167), T168, T169, .(T166, X322)) → U9_ggga(T166, T167, T168, T169, X322, appL_in_ggga(T167, T168, T169, X322))
appL_in_ggga([], T178, T179, .(T178, T179)) → appL_out_ggga([], T178, T179, .(T178, T179))
U9_ggga(T166, T167, T168, T169, X322, appL_out_ggga(T167, T168, T169, X322)) → appL_out_ggga(.(T166, T167), T168, T169, .(T166, X322))
U28_gagga(T132, T143, T138, T126, X246, appL_out_ggga(T138, T126, T143, X246)) → pS_out_gagga(T132, T143, T138, T126, X246)
U26_gagaga(T131, T138, T132, X245, T126, X246, pS_out_gagga(T132, X245, T138, T126, X246)) → pR_out_gagaga(T131, T138, T132, X245, T126, X246)
U24_ggaaaaa(T126, T127, T131, T132, X244, X245, X246, pR_out_gagaga(T131, X244, T132, X245, T126, X246)) → pK_out_ggaaaaa(T126, T127, T131, T132, X244, X245, X246)
U8_ga(T126, T127, X246, pK_out_ggaaaaa(T126, T127, X242, X243, X244, X245, X246)) → qsJ_out_ga(.(T126, T127), X246)
U33_gaggaga(T244, T251, T235, T245, X10, T234, T9, qsJ_out_ga(T244, T251)) → U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_in_gagga(.(T235, T245), X10, T251, T234, T9))
pQ_in_gagga(T45, T115, T110, T22, T9) → U21_gagga(T45, T115, T110, T22, T9, qsJ_in_ga(T45, T115))
U21_gagga(T45, T115, T110, T22, T9, qsJ_out_ga(T45, T115)) → U22_gagga(T45, T115, T110, T22, T9, appM_in_ggga(T110, T22, T115, T9))
appM_in_ggga(.(T205, T206), T207, T208, .(T205, T210)) → U10_ggga(T205, T206, T207, T208, T210, appM_in_ggga(T206, T207, T208, T210))
appM_in_ggga([], T220, T221, .(T220, T221)) → appM_out_ggga([], T220, T221, .(T220, T221))
U10_ggga(T205, T206, T207, T208, T210, appM_out_ggga(T206, T207, T208, T210)) → appM_out_ggga(.(T205, T206), T207, T208, .(T205, T210))
U22_gagga(T45, T115, T110, T22, T9, appM_out_ggga(T110, T22, T115, T9)) → pQ_out_gagga(T45, T115, T110, T22, T9)
U34_gaggaga(T244, T251, T235, T245, X10, T234, T9, pQ_out_gagga(.(T235, T245), X10, T251, T234, T9)) → pU_out_gaggaga(T244, T251, T235, T245, X10, T234, T9)
U32_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9, pU_out_gaggaga(T244, X9, T235, T245, X10, T234, T9)) → pT_out_ggaaagaa(T234, T236, T244, T245, X9, T235, X10, T9)
U30_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9, pT_out_ggaaagaa(T234, T236, X395, X396, X9, T235, X10, T9)) → pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)
U2_ga(T234, T235, T236, T9, pC_out_gggaaaaa(T234, T235, T236, X395, X396, X9, X10, T9)) → qsA_out_ga(.(T234, .(T235, T236)), T9)
qsA_in_ga(.(T260, []), T9) → U3_ga(T260, T9, pD_in_aaga(X9, X10, T260, T9))
pD_in_aaga(T261, X10, T260, T9) → U35_aaga(T261, X10, T260, T9, qsN_in_a(T261))
qsN_in_a([]) → qsN_out_a([])
U35_aaga(T261, X10, T260, T9, qsN_out_a(T261)) → U36_aaga(T261, X10, T260, T9, pV_in_agga(X10, T261, T260, T9))
pV_in_agga(T262, T261, T260, T9) → U37_agga(T262, T261, T260, T9, qsN_in_a(T262))
U37_agga(T262, T261, T260, T9, qsN_out_a(T262)) → U38_agga(T262, T261, T260, T9, appM_in_ggga(T261, T260, T262, T9))
U38_agga(T262, T261, T260, T9, appM_out_ggga(T261, T260, T262, T9)) → pV_out_agga(T262, T261, T260, T9)
U36_aaga(T261, X10, T260, T9, pV_out_agga(X10, T261, T260, T9)) → pD_out_aaga(T261, X10, T260, T9)
U3_ga(T260, T9, pD_out_aaga(X9, X10, T260, T9)) → qsA_out_ga(.(T260, []), T9)
qsA_in_ga([], []) → qsA_out_ga([], [])
U15_ggagaga(T23, T44, T110, T45, X10, T22, T9, qsA_out_ga(.(T23, T44), T110)) → U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_in_gagga(T45, X10, T110, T22, T9))
U16_ggagaga(T23, T44, T110, T45, X10, T22, T9, pQ_out_gagga(T45, X10, T110, T22, T9)) → pP_out_ggagaga(T23, T44, T110, T45, X10, T22, T9)
U14_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9, pP_out_ggagaga(T23, T44, X9, T45, X10, T22, T9)) → pO_out_ggaagaaa(T22, T24, T44, T45, T23, X9, X10, T9)
U12_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9, pO_out_ggaagaaa(T22, T24, X44, X45, T23, X9, X10, T9)) → pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, T9)
U1_ga(T22, T23, T24, T9, pB_out_gggaaaaa(T22, T23, T24, X44, X45, X9, X10, 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)
.(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)
U11_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_gggaaaaa(x1, x2, x3, x9)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
U12_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U12_gggaaaaa(x1, x2, x3, x9)
pO_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_in_ggaagaaa(x1, x2, x5)
U13_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
pG_out_gggaa(x1, x2, x3, x4, x5)  =  pG_out_gggaa(x1, x2, x3, x4, x5)
U14_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_ggaagaaa(x1, x2, x3, x4, x5, x9)
pP_in_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_in_ggagaga(x1, x2, x4, x6)
U15_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_ggagaga(x1, x2, x4, x6, x8)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_in_gggaaaaa(x1, x2, x3)
U29_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U29_gggaaaaa(x1, x2, x3, x9)
U30_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U30_gggaaaaa(x1, x2, x3, x9)
pT_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_in_ggaaagaa(x1, x2, x6)
U31_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U31_ggaaagaa(x1, x2, x6, x9)
U32_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U32_ggaaagaa(x1, x2, x3, x4, x6, x9)
pU_in_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_in_gaggaga(x1, x3, x4, x6)
U33_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U33_gaggaga(x1, x3, x4, x6, x8)
qsJ_in_ga(x1, x2)  =  qsJ_in_ga(x1)
U8_ga(x1, x2, x3, x4)  =  U8_ga(x1, x2, x4)
pK_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_in_ggaaaaa(x1, x2)
U23_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U23_ggaaaaa(x1, x2, x8)
U24_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U24_ggaaaaa(x1, x2, x3, x4, x8)
pR_in_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_in_gagaga(x1, x3, x5)
U25_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U25_gagaga(x1, x3, x5, x7)
qsJ_out_ga(x1, x2)  =  qsJ_out_ga(x1, x2)
U26_gagaga(x1, x2, x3, x4, x5, x6, x7)  =  U26_gagaga(x1, x2, x3, x5, x7)
pS_in_gagga(x1, x2, x3, x4, x5)  =  pS_in_gagga(x1, x3, x4)
U27_gagga(x1, x2, x3, x4, x5, x6)  =  U27_gagga(x1, x3, x4, x6)
U28_gagga(x1, x2, x3, x4, x5, x6)  =  U28_gagga(x1, x2, x3, x4, x6)
appL_in_ggga(x1, x2, x3, x4)  =  appL_in_ggga(x1, x2, x3)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
appL_out_ggga(x1, x2, x3, x4)  =  appL_out_ggga(x1, x2, x3, x4)
pS_out_gagga(x1, x2, x3, x4, x5)  =  pS_out_gagga(x1, x2, x3, x4, x5)
pR_out_gagaga(x1, x2, x3, x4, x5, x6)  =  pR_out_gagaga(x1, x2, x3, x4, x5, x6)
pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pK_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U34_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_gaggaga(x1, x2, x3, x4, x6, x8)
pQ_in_gagga(x1, x2, x3, x4, x5)  =  pQ_in_gagga(x1, x3, x4)
U21_gagga(x1, x2, x3, x4, x5, x6)  =  U21_gagga(x1, x3, x4, x6)
U22_gagga(x1, x2, x3, x4, x5, x6)  =  U22_gagga(x1, x2, x3, x4, x6)
appM_in_ggga(x1, x2, x3, x4)  =  appM_in_ggga(x1, x2, x3)
U10_ggga(x1, x2, x3, x4, x5, x6)  =  U10_ggga(x1, x2, x3, x4, x6)
appM_out_ggga(x1, x2, x3, x4)  =  appM_out_ggga(x1, x2, x3, x4)
pQ_out_gagga(x1, x2, x3, x4, x5)  =  pQ_out_gagga(x1, x2, x3, x4, x5)
pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)  =  pU_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pT_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pC_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
qsA_out_ga(x1, x2)  =  qsA_out_ga(x1, x2)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
pD_in_aaga(x1, x2, x3, x4)  =  pD_in_aaga(x3)
U35_aaga(x1, x2, x3, x4, x5)  =  U35_aaga(x3, x5)
qsN_in_a(x1)  =  qsN_in_a
qsN_out_a(x1)  =  qsN_out_a(x1)
U36_aaga(x1, x2, x3, x4, x5)  =  U36_aaga(x1, x3, x5)
pV_in_agga(x1, x2, x3, x4)  =  pV_in_agga(x2, x3)
U37_agga(x1, x2, x3, x4, x5)  =  U37_agga(x2, x3, x5)
U38_agga(x1, x2, x3, x4, x5)  =  U38_agga(x1, x2, x3, x5)
pV_out_agga(x1, x2, x3, x4)  =  pV_out_agga(x1, x2, x3, x4)
pD_out_aaga(x1, x2, x3, x4)  =  pD_out_aaga(x1, x2, x3, x4)
U16_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_ggagaga(x1, x2, x3, x4, x6, x8)
pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)  =  pP_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pO_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  pO_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)
U11_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_GGGAAAAA(x1, x2, x3, x9)
PO_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PO_IN_GGAAGAAA(x1, x2, x5)
U13_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_GGAAGAAA(x1, x2, x5, x9)
PP_IN_GGAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PP_IN_GGAGAGA(x1, x2, x4, x6)

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

(52) UsableRulesProof (EQUIVALENT transformation)

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

(53) 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, X44, X45, X9, X10, T9)
PB_IN_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9) → U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_in_gg(T22, T23))
U11_GGGAAAAA(T22, T23, T24, X44, X45, X9, X10, T9, gtE_out_gg(T22, T23)) → PO_IN_GGAAGAAA(T22, T24, X44, X45, T23, X9, X10, T9)
PO_IN_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9) → U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_in_ggaa(T22, T24, T44, T45))
U13_GGAAGAAA(T22, T24, T44, T45, T23, X9, X10, T9, partF_out_ggaa(T22, T24, T44, T45)) → PP_IN_GGAGAGA(T23, T44, X9, T45, X10, T22, T9)
PP_IN_GGAGAGA(T23, T44, T110, T45, X10, T22, T9) → QSA_IN_GA(.(T23, T44), T110)

The TRS R consists of the following rules:

gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
partF_in_ggaa(T61, .(T62, T63), .(T62, X109), X110) → U5_ggaa(T61, T62, T63, X109, X110, pG_in_gggaa(T61, T62, T63, X109, X110))
partF_in_ggaa(T83, .(T84, T85), X155, .(T84, X156)) → U6_ggaa(T83, T84, T85, X155, X156, pH_in_gggaa(T83, T84, T85, X155, X156))
partF_in_ggaa(T107, [], [], []) → partF_out_ggaa(T107, [], [], [])
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U5_ggaa(T61, T62, T63, X109, X110, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U6_ggaa(T83, T84, T85, X155, X156, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
pG_in_gggaa(T61, T62, T63, X109, X110) → U17_gggaa(T61, T62, T63, X109, X110, gtE_in_gg(T61, T62))
pH_in_gggaa(T83, T84, T85, X155, X156) → U19_gggaa(T83, T84, T85, X155, X156, leI_in_gg(T83, T84))
U17_gggaa(T61, T62, T63, X109, X110, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, X109, X110, partF_in_ggaa(T61, T63, X109, X110))
U19_gggaa(T83, T84, T85, X155, X156, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, X155, X156, partF_in_ggaa(T83, T85, X155, X156))
U18_gggaa(T61, T62, T63, X109, X110, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U20_gggaa(T83, T84, T85, X155, X156, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
gtE_in_gg(x1, x2)  =  gtE_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
gtE_out_gg(x1, x2)  =  gtE_out_gg(x1, x2)
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)
U17_gggaa(x1, x2, x3, x4, x5, x6)  =  U17_gggaa(x1, x2, x3, x6)
U18_gggaa(x1, x2, x3, x4, x5, x6)  =  U18_gggaa(x1, x2, x3, x6)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
pH_in_gggaa(x1, x2, x3, x4, x5)  =  pH_in_gggaa(x1, x2, x3)
U19_gggaa(x1, x2, x3, x4, x5, x6)  =  U19_gggaa(x1, x2, x3, x6)
leI_in_gg(x1, x2)  =  leI_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
leI_out_gg(x1, x2)  =  leI_out_gg(x1, x2)
U20_gggaa(x1, x2, x3, x4, x5, x6)  =  U20_gggaa(x1, x2, x3, x6)
[]  =  []
partF_out_ggaa(x1, x2, x3, x4)  =  partF_out_ggaa(x1, x2, x3, x4)
pH_out_gggaa(x1, x2, x3, x4, x5)  =  pH_out_gggaa(x1, x2, x3, x4, x5)
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)
U11_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U11_GGGAAAAA(x1, x2, x3, x9)
PO_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  PO_IN_GGAAGAAA(x1, x2, x5)
U13_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U13_GGAAGAAA(x1, x2, x5, x9)
PP_IN_GGAGAGA(x1, x2, x3, x4, x5, x6, x7)  =  PP_IN_GGAGAGA(x1, x2, x4, x6)

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

(54) PiDPToQDPProof (SOUND transformation)

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

(55) 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) → U11_GGGAAAAA(T22, T23, T24, gtE_in_gg(T22, T23))
U11_GGGAAAAA(T22, T23, T24, gtE_out_gg(T22, T23)) → PO_IN_GGAAGAAA(T22, T24, T23)
PO_IN_GGAAGAAA(T22, T24, T23) → U13_GGAAGAAA(T22, T24, T23, partF_in_ggaa(T22, T24))
U13_GGAAGAAA(T22, T24, T23, partF_out_ggaa(T22, T24, T44, T45)) → PP_IN_GGAGAGA(T23, T44, T45, T22)
PP_IN_GGAGAGA(T23, T44, T45, T22) → QSA_IN_GA(.(T23, T44))

The TRS R consists of the following rules:

gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
partF_in_ggaa(T61, .(T62, T63)) → U5_ggaa(T61, T62, T63, pG_in_gggaa(T61, T62, T63))
partF_in_ggaa(T83, .(T84, T85)) → U6_ggaa(T83, T84, T85, pH_in_gggaa(T83, T84, T85))
partF_in_ggaa(T107, []) → partF_out_ggaa(T107, [], [], [])
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U5_ggaa(T61, T62, T63, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U6_ggaa(T83, T84, T85, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
pG_in_gggaa(T61, T62, T63) → U17_gggaa(T61, T62, T63, gtE_in_gg(T61, T62))
pH_in_gggaa(T83, T84, T85) → U19_gggaa(T83, T84, T85, leI_in_gg(T83, T84))
U17_gggaa(T61, T62, T63, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, partF_in_ggaa(T61, T63))
U19_gggaa(T83, T84, T85, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, partF_in_ggaa(T83, T85))
U18_gggaa(T61, T62, T63, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U20_gggaa(T83, T84, T85, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))

The set Q consists of the following terms:

gtE_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)
pH_in_gggaa(x0, x1, x2)
U17_gggaa(x0, x1, x2, x3)
U19_gggaa(x0, x1, x2, x3)
U18_gggaa(x0, x1, x2, x3)
leI_in_gg(x0, x1)
U20_gggaa(x0, x1, x2, x3)
U7_gg(x0, x1, x2)

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

(56) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U11_GGGAAAAA(T22, T23, T24, gtE_out_gg(T22, T23)) → PO_IN_GGAAGAAA(T22, T24, T23)

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

POL(.(x₁, x₂)) = x₁   
POL(0) = 0   
POL(PB_IN_GGGAAAAA(x₁, x₂, x₃)) = x₁   
POL(PO_IN_GGAAGAAA(x₁, x₂, x₃)) = x₃   
POL(PP_IN_GGAGAGA(x₁, x₂, x₃, x₄)) = x₁   
POL(QSA_IN_GA(x₁)) = x₁   
POL(U11_GGGAAAAA(x₁, x₂, x₃, x₄)) = x₄   
POL(U13_GGAAGAAA(x₁, x₂, x₃, x₄)) = x₃   
POL(U17_gggaa(x₁, x₂, x₃, x₄)) = 0   
POL(U18_gggaa(x₁, x₂, x₃, x₄)) = 0   
POL(U19_gggaa(x₁, x₂, x₃, x₄)) = 0   
POL(U20_gggaa(x₁, x₂, x₃, x₄)) = 0   
POL(U4_gg(x₁, x₂, x₃)) = 1 + x₃   
POL(U5_ggaa(x₁, x₂, x₃, x₄)) = 0   
POL(U6_ggaa(x₁, x₂, x₃, x₄)) = 0   
POL(U7_gg(x₁, x₂, x₃)) = 1 + x₂   
POL([]) = 0   
POL(gtE_in_gg(x₁, x₂)) = x₁   
POL(gtE_out_gg(x₁, x₂)) = 1 + x₂   
POL(leI_in_gg(x₁, x₂)) = 1 + x₁ + x₂   
POL(leI_out_gg(x₁, x₂)) = x₂   
POL(pG_in_gggaa(x₁, x₂, x₃)) = 0   
POL(pG_out_gggaa(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(pH_in_gggaa(x₁, x₂, x₃)) = 0   
POL(pH_out_gggaa(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(partF_in_ggaa(x₁, x₂)) = 0   
POL(partF_out_ggaa(x₁, x₂, x₃, x₄)) = 0   
POL(s(x₁)) = 1 + x₁   

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

gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))

(57) 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) → U11_GGGAAAAA(T22, T23, T24, gtE_in_gg(T22, T23))
PO_IN_GGAAGAAA(T22, T24, T23) → U13_GGAAGAAA(T22, T24, T23, partF_in_ggaa(T22, T24))
U13_GGAAGAAA(T22, T24, T23, partF_out_ggaa(T22, T24, T44, T45)) → PP_IN_GGAGAGA(T23, T44, T45, T22)
PP_IN_GGAGAGA(T23, T44, T45, T22) → QSA_IN_GA(.(T23, T44))

The TRS R consists of the following rules:

gtE_in_gg(s(T37), s(T38)) → U4_gg(T37, T38, gtE_in_gg(T37, T38))
gtE_in_gg(s(0), 0) → gtE_out_gg(s(0), 0)
partF_in_ggaa(T61, .(T62, T63)) → U5_ggaa(T61, T62, T63, pG_in_gggaa(T61, T62, T63))
partF_in_ggaa(T83, .(T84, T85)) → U6_ggaa(T83, T84, T85, pH_in_gggaa(T83, T84, T85))
partF_in_ggaa(T107, []) → partF_out_ggaa(T107, [], [], [])
U4_gg(T37, T38, gtE_out_gg(T37, T38)) → gtE_out_gg(s(T37), s(T38))
U5_ggaa(T61, T62, T63, pG_out_gggaa(T61, T62, T63, X109, X110)) → partF_out_ggaa(T61, .(T62, T63), .(T62, X109), X110)
U6_ggaa(T83, T84, T85, pH_out_gggaa(T83, T84, T85, X155, X156)) → partF_out_ggaa(T83, .(T84, T85), X155, .(T84, X156))
pG_in_gggaa(T61, T62, T63) → U17_gggaa(T61, T62, T63, gtE_in_gg(T61, T62))
pH_in_gggaa(T83, T84, T85) → U19_gggaa(T83, T84, T85, leI_in_gg(T83, T84))
U17_gggaa(T61, T62, T63, gtE_out_gg(T61, T62)) → U18_gggaa(T61, T62, T63, partF_in_ggaa(T61, T63))
U19_gggaa(T83, T84, T85, leI_out_gg(T83, T84)) → U20_gggaa(T83, T84, T85, partF_in_ggaa(T83, T85))
U18_gggaa(T61, T62, T63, partF_out_ggaa(T61, T63, X109, X110)) → pG_out_gggaa(T61, T62, T63, X109, X110)
leI_in_gg(s(T98), s(T99)) → U7_gg(T98, T99, leI_in_gg(T98, T99))
leI_in_gg(0, s(0)) → leI_out_gg(0, s(0))
leI_in_gg(0, 0) → leI_out_gg(0, 0)
U20_gggaa(T83, T84, T85, partF_out_ggaa(T83, T85, X155, X156)) → pH_out_gggaa(T83, T84, T85, X155, X156)
U7_gg(T98, T99, leI_out_gg(T98, T99)) → leI_out_gg(s(T98), s(T99))

The set Q consists of the following terms:

gtE_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)
pH_in_gggaa(x0, x1, x2)
U17_gggaa(x0, x1, x2, x3)
U19_gggaa(x0, x1, x2, x3)
U18_gggaa(x0, x1, x2, x3)
leI_in_gg(x0, x1)
U20_gggaa(x0, x1, x2, x3)
U7_gg(x0, x1, x2)

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

(58) DependencyGraphProof (EQUIVALENT transformation)

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

(59) TRUE