Left Termination proof of /tmp/tmpaJrKAb/qsort.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 319 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 1198 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 49 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇒, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔, 0 ms)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇔, 0 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔, 0 ms)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇔, 0 ms)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔, 0 ms)

        ↳27 YES

    ↳28 PiDP

        ↳29 UsableRulesProof (⇔, 0 ms)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇒, 5 ms)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔, 0 ms)

        ↳34 YES

    ↳35 PiDP

        ↳36 UsableRulesProof (⇔, 0 ms)

        ↳37 PiDP

        ↳38 PiDPToQDPProof (⇒, 0 ms)

        ↳39 QDP

        ↳40 QDPOrderProof (⇔, 294 ms)

        ↳41 QDP

        ↳42 DependencyGraphProof (⇔, 0 ms)

        ↳43 TRUE

    ↳44 PiDP

        ↳45 UsableRulesProof (⇔, 0 ms)

        ↳46 PiDP

        ↳47 PiDPToQDPProof (⇒, 0 ms)

        ↳48 QDP

        ↳49 QDPSizeChangeProof (⇔, 0 ms)

        ↳50 YES

    ↳51 PiDP

        ↳52 UsableRulesProof (⇔, 0 ms)

        ↳53 PiDP

        ↳54 PiDPToQDPProof (⇒, 0 ms)

        ↳55 QDP

        ↳56 QDPOrderProof (⇔, 310 ms)

        ↳57 QDP

        ↳58 DependencyGraphProof (⇔, 0 ms)

        ↳59 TRUE

(0) Obligation:

Clauses:

qs([], []).
qs(cons(X, L), S) :- ','(split(L, X, L1, L2), ','(qs(L1, S1), ','(qs(L2, S2), append(S1, cons(X, S2), S)))).
append([], L, L).
append(cons(X, L1), L2, cons(X, L3)) :- append(L1, L2, L3).
split([], X, [], []).
split(cons(X, L), Y, cons(X, L1), L2) :- ','(less(X, Y), split(L, Y, L1, L2)).
split(cons(X, L), Y, L1, cons(X, L2)) :- ','(geq(X, Y), split(L, Y, L1, L2)).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).
geq(X, X).
geq(s(X), 0).
geq(s(X), s(Y)) :- geq(X, Y).

Query: qs(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: qs(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: qs(T1, T2), SCOPE: 1, CLAUSE: 0 | qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | qs([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: qs(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nqs(T1, T2) !=? qs([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: qs([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(split(T7, T6, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(T6, X13), T9))))\n\nT6 is ground\nT7 is ground\nX10 is free\nX11 is free\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(split(T7, T6, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(T6, X13), T9)))), SCOPE: 2, CLAUSE: 4 | ','(split(T7, T6, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(T6, X13), T9)))), SCOPE: 2, CLAUSE: 5 | ','(split(T7, T6, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(T6, X13), T9)))), SCOPE: 2, CLAUSE: 6\n\nT6 is ground\nT7 is ground\nX10 is free\nX11 is free\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(split(T7, T6, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(T6, X13), T9)))), SCOPE: 2, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX10 is free\nX11 is free\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(split(T7, T6, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(T6, X13), T9)))), SCOPE: 2, CLAUSE: 5 | ','(split(T7, T6, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(T6, X13), T9)))), SCOPE: 2, CLAUSE: 6\n\nT6 is ground\nT7 is ground\nX10 is free\nX11 is free\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="B: ','(qs([], X12), ','(qs([], X13), append(X12, cons(T14, X13), T9)))\n\nT14 is ground\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="E: qs([], X12)\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="O: ','(qs([], X13), append(T19, cons(T14, X13), T9))\n\nT14 is ground\nT19 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: qs([], X12), SCOPE: 3, CLAUSE: 0 | qs([], X12), SCOPE: 3, CLAUSE: 1\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: qs([], X12), SCOPE: 3, CLAUSE: 0\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: qs([], X12), SCOPE: 3, CLAUSE: 1\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: qs([], X13)\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="F: append(T19, cons(T14, T24), T9)\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: append(T19, cons(T14, T24), T9), SCOPE: 4, CLAUSE: 2 | append(T19, cons(T14, T24), T9), SCOPE: 4, CLAUSE: 3\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: append(T19, cons(T14, T24), T9), SCOPE: 4, CLAUSE: 2\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: append(T19, cons(T14, T24), T9), SCOPE: 4, CLAUSE: 3\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: append(T50, cons(T51, T52), T54)\n\nT50 is ground\nT51 is ground\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(split(T7, T6, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(T6, X13), T9)))), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nT7 is ground\nX10 is free\nX11 is free\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(split(T7, T6, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(T6, X13), T9)))), SCOPE: 2, CLAUSE: 6\n\nT6 is ground\nT7 is ground\nX10 is free\nX11 is free\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="C: ','(','(less(T71, T73), split(T72, T73, X84, X85)), ','(qs(cons(T71, X84), X12), ','(qs(X85, X13), append(X12, cons(T73, X13), T9))))\n\nT71 is ground\nT72 is ground\nT73 is ground\nX12 is free\nX13 is free\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="G: less(T71, T73)\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="P: ','(split(T72, T73, X84, X85), ','(qs(cons(T71, X84), X12), ','(qs(X85, X13), append(X12, cons(T73, X13), T9))))\n\nT71 is ground\nT72 is ground\nT73 is ground\nX12 is free\nX13 is free\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: less(T71, T73), SCOPE: 5, CLAUSE: 7 | less(T71, T73), SCOPE: 5, CLAUSE: 8\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: less(T71, T73), SCOPE: 5, CLAUSE: 7\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: less(T71, T73), SCOPE: 5, CLAUSE: 8\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: less(T87, T88)\n\nT87 is ground\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="H: split(T72, T73, X84, X85)\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="Q: ','(qs(cons(T71, T93), X12), ','(qs(T94, X13), append(X12, cons(T73, X13), T9)))\n\nT71 is ground\nT73 is ground\nT93 is ground\nT94 is ground\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 4 | split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 5 | split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 6\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 4\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 5 | split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 6\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="143: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
144 [label="144: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
148 [label="148: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 5\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 6\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
158 [label="I: ','(less(T114, T116), split(T115, T116, X145, X146))\n\nT114 is ground\nT115 is ground\nT116 is ground\nX145 is free\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: less(T114, T116)\n\nT114 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: split(T115, T116, X145, X146)\n\nT115 is ground\nT116 is ground\nX145 is free\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="J: ','(geq(T129, T131), split(T130, T131, X172, X173))\n\nT129 is ground\nT130 is ground\nT131 is ground\nX172 is free\nX173 is free", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="K: geq(T129, T131)\n\nT129 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: split(T130, T131, X172, X173)\n\nT130 is ground\nT131 is ground\nX172 is free\nX173 is free", fontsize=16, style=filled, fillcolor=lightyellow];
205 [label="205: geq(T129, T131), SCOPE: 7, CLAUSE: 9 | geq(T129, T131), SCOPE: 7, CLAUSE: 10 | geq(T129, T131), SCOPE: 7, CLAUSE: 11\n\nT129 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: geq(T129, T131), SCOPE: 7, CLAUSE: 9\n\nT129 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
207 [label="207: geq(T129, T131), SCOPE: 7, CLAUSE: 10 | geq(T129, T131), SCOPE: 7, CLAUSE: 11\n\nT129 is ground\nT131 is ground", 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: geq(T129, T131), SCOPE: 7, CLAUSE: 10\n\nT129 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
214 [label="214: geq(T129, T131), SCOPE: 7, CLAUSE: 11\n\nT129 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
217 [label="217: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
218 [label="218: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
219 [label="219: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: geq(T150, T151)\n\nT150 is ground\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
235 [label="235: qs(cons(T71, T93), X12)\n\nT71 is ground\nT93 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
236 [label="R: ','(qs(T94, X13), append(T156, cons(T73, X13), T9))\n\nT73 is ground\nT94 is ground\nT156 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
237 [label="L: qs(T94, X13)\n\nT94 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
238 [label="238: append(T156, cons(T73, T157), T9)\n\nT73 is ground\nT156 is ground\nT157 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
239 [label="239: qs(T94, X13), SCOPE: 8, CLAUSE: 0 | qs(T94, X13), SCOPE: 8, CLAUSE: 1\n\nT94 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
240 [label="240: qs(T94, X13), SCOPE: 8, CLAUSE: 0\n\nT94 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
241 [label="241: qs(T94, X13), SCOPE: 8, CLAUSE: 1\n\nT94 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
244 [label="244: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
245 [label="245: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
246 [label="246: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
249 [label="M: ','(split(T163, T162, X213, X214), ','(qs(X213, X215), ','(qs(X214, X216), append(X215, cons(T162, X216), X217))))\n\nT162 is ground\nT163 is ground\nX217 is free\nX213 is free\nX214 is free\nX215 is free\nX216 is free", fontsize=16, style=filled, fillcolor=lightyellow];
250 [label="250: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
257 [label="257: split(T163, T162, X213, X214)\n\nT162 is ground\nT163 is ground\nX213 is free\nX214 is free", fontsize=16, style=filled, fillcolor=lightyellow];
258 [label="S: ','(qs(T166, X215), ','(qs(T167, X216), append(X215, cons(T162, X216), X217)))\n\nT162 is ground\nT166 is ground\nT167 is ground\nX217 is free\nX215 is free\nX216 is free", fontsize=16, style=filled, fillcolor=lightyellow];
261 [label="261: qs(T166, X215)\n\nT166 is ground\nX215 is free", fontsize=16, style=filled, fillcolor=lightyellow];
262 [label="T: ','(qs(T167, X216), append(T170, cons(T162, X216), X217))\n\nT162 is ground\nT167 is ground\nT170 is ground\nX217 is free\nX216 is free", fontsize=16, style=filled, fillcolor=lightyellow];
265 [label="265: qs(T167, X216)\n\nT167 is ground\nX216 is free", fontsize=16, style=filled, fillcolor=lightyellow];
266 [label="N: append(T170, cons(T162, T171), X217)\n\nT162 is ground\nT170 is ground\nT171 is ground\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
269 [label="269: append(T170, cons(T162, T171), X217), SCOPE: 9, CLAUSE: 2 | append(T170, cons(T162, T171), X217), SCOPE: 9, CLAUSE: 3\n\nT162 is ground\nT170 is ground\nT171 is ground\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
270 [label="270: append(T170, cons(T162, T171), X217), SCOPE: 9, CLAUSE: 2\n\nT162 is ground\nT170 is ground\nT171 is ground\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
271 [label="271: append(T170, cons(T162, T171), X217), SCOPE: 9, CLAUSE: 3\n\nT162 is ground\nT170 is ground\nT171 is ground\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
273 [label="273: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
275 [label="275: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
276 [label="276: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
278 [label="278: append(T195, cons(T196, T197), X243)\n\nT195 is ground\nT196 is ground\nT197 is ground\nX243 is free", fontsize=16, style=filled, fillcolor=lightyellow];
279 [label="279: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
285 [label="D: ','(','(geq(T212, T214), split(T213, T214, X267, X268)), ','(qs(X267, X12), ','(qs(cons(T212, X268), X13), append(X12, cons(T214, X13), T9))))\n\nT212 is ground\nT213 is ground\nT214 is ground\nX12 is free\nX13 is free\nX267 is free\nX268 is free", fontsize=16, style=filled, fillcolor=lightyellow];
286 [label="286: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
287 [label="287: geq(T212, T214)\n\nT212 is ground\nT214 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
288 [label="U: ','(split(T213, T214, X267, X268), ','(qs(X267, X12), ','(qs(cons(T212, X268), X13), append(X12, cons(T214, X13), T9))))\n\nT212 is ground\nT213 is ground\nT214 is ground\nX12 is free\nX13 is free\nX267 is free\nX268 is free", fontsize=16, style=filled, fillcolor=lightyellow];
291 [label="291: split(T213, T214, X267, X268)\n\nT213 is ground\nT214 is ground\nX267 is free\nX268 is free", fontsize=16, style=filled, fillcolor=lightyellow];
292 [label="V: ','(qs(T221, X12), ','(qs(cons(T212, T222), X13), append(X12, cons(T214, X13), T9)))\n\nT212 is ground\nT214 is ground\nT221 is ground\nT222 is ground\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
293 [label="293: qs(T221, X12)\n\nT221 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
294 [label="294: ','(qs(cons(T212, T222), X13), append(T225, cons(T214, X13), T9))\n\nT212 is ground\nT214 is ground\nT222 is ground\nT225 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
295 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 295 [arrowhead = none ];
295 -> 4;

296 [label="EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 296 [arrowhead = none ];
296 -> 10;

297 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 297 [arrowhead = none ];
297 -> 13;

298 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 298 [arrowhead = none ];
298 -> 16;

299 [label="EVAL with clause\nqs(cons(X7, X8), X9) :- ','(split(X8, X7, X10, X11), ','(qs(X10, X12), ','(qs(X11, X13), append(X12, cons(X7, X13), X9)))).\nand substitutionX7 -> T6,\nX8 -> T7,\nT1 -> cons(T6, T7),\nT2 -> T9,\nX9 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 299 [arrowhead = none ];
299 -> 29;

300 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 300 [arrowhead = none ];
300 -> 30;

301 [label="BACKTRACK\nfor clause: qs(cons(X, L), S) :- ','(split(L, X, L1, L2), ','(qs(L1, S1), ','(qs(L2, S2), append(S1, cons(X, S2), S))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 301 [arrowhead = none ];
301 -> 18;

302 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 302 [arrowhead = none ];
302 -> 33;

303 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 303 [arrowhead = none ];
303 -> 34;

304 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 304 [arrowhead = none ];
304 -> 35;

305 [label="EVAL with clause\nsplit([], X18, [], []).\nand substitutionT7 -> [],\nT6 -> T14,\nX18 -> T14,\nX10 -> [],\nX11 -> []", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 305 [arrowhead = none ];
305 -> 37;

306 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 306 [arrowhead = none ];
306 -> 40;

307 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 307 [arrowhead = none ];
307 -> 92;

308 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 308 [arrowhead = none ];
308 -> 94;

309 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
37 -> 309 [arrowhead = none ];
309 -> 44;

310 [label="SPLIT 2\nnew knowledge:\nT19 is ground\nreplacements:X12 -> T19", fontsize=14, style = filled, fillcolor = violet];
37 -> 310 [arrowhead = none ];
310 -> 45;

311 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 311 [arrowhead = none ];
311 -> 46;

312 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
45 -> 312 [arrowhead = none ];
312 -> 69;

313 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nreplacements:X13 -> T24", fontsize=14, style = filled, fillcolor = violet];
45 -> 313 [arrowhead = none ];
313 -> 70;

314 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 314 [arrowhead = none ];
314 -> 59;

315 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 315 [arrowhead = none ];
315 -> 60;

316 [label="ONLY EVAL with clause\nqs([], []).\nand substitutionX12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 316 [arrowhead = none ];
316 -> 61;

317 [label="BACKTRACK\nfor clause: qs(cons(X, L), S) :- ','(split(L, X, L1, L2), ','(qs(L1, S1), ','(qs(L2, S2), append(S1, cons(X, S2), S))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 317 [arrowhead = none ];
317 -> 63;

318 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 318 [arrowhead = none ];
318 -> 62;

319 [label="INSTANCE with matching:\nX12 -> X13", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 319 [arrowhead = none , style=dashed];
319 -> 44[style=dashed];

320 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 320 [arrowhead = none ];
320 -> 74;

321 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 321 [arrowhead = none ];
321 -> 79;

322 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
74 -> 322 [arrowhead = none ];
322 -> 80;

323 [label="EVAL with clause\nappend([], X36, X36).\nand substitutionT19 -> [],\nT14 -> T37,\nT24 -> T38,\nX36 -> cons(T37, T38),\nT9 -> cons(T37, T38)", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 323 [arrowhead = none ];
323 -> 81;

324 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 324 [arrowhead = none ];
324 -> 82;

325 [label="EVAL with clause\nappend(cons(X45, X46), X47, cons(X45, X48)) :- append(X46, X47, X48).\nand substitutionX45 -> T49,\nX46 -> T50,\nT19 -> cons(T49, T50),\nT14 -> T51,\nT24 -> T52,\nX47 -> cons(T51, T52),\nX48 -> T54,\nT9 -> cons(T49, T54),\nT53 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 325 [arrowhead = none ];
325 -> 85;

326 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 326 [arrowhead = none ];
326 -> 86;

327 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 327 [arrowhead = none ];
327 -> 83;

328 [label="INSTANCE with matching:\nT19 -> T50\nT14 -> T51\nT24 -> T52\nT9 -> T54", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 328 [arrowhead = none , style=dashed];
328 -> 70[style=dashed];

329 [label="EVAL with clause\nsplit(cons(X79, X80), X81, cons(X79, X82), X83) :- ','(less(X79, X81), split(X80, X81, X82, X83)).\nand substitutionX79 -> T71,\nX80 -> T72,\nT7 -> cons(T71, T72),\nT6 -> T73,\nX81 -> T73,\nX82 -> X84,\nX10 -> cons(T71, X84),\nX11 -> X85,\nX83 -> X85", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 329 [arrowhead = none ];
329 -> 106;

330 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 330 [arrowhead = none ];
330 -> 107;

331 [label="EVAL with clause\nsplit(cons(X262, X263), X264, X265, cons(X262, X266)) :- ','(geq(X262, X264), split(X263, X264, X265, X266)).\nand substitutionX262 -> T212,\nX263 -> T213,\nT7 -> cons(T212, T213),\nT6 -> T214,\nX264 -> T214,\nX10 -> X267,\nX265 -> X267,\nX266 -> X268,\nX11 -> cons(T212, X268)", fontsize=14, style = filled, fillcolor = lightblue];
94 -> 331 [arrowhead = none ];
331 -> 285;

332 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
94 -> 332 [arrowhead = none ];
332 -> 286;

333 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
106 -> 333 [arrowhead = none ];
333 -> 111;

334 [label="SPLIT 2\nnew knowledge:\nT71 is ground\nT73 is ground", fontsize=14, style = filled, fillcolor = violet];
106 -> 334 [arrowhead = none ];
334 -> 113;

335 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
111 -> 335 [arrowhead = none ];
335 -> 115;

336 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
113 -> 336 [arrowhead = none ];
336 -> 133;

337 [label="SPLIT 2\nnew knowledge:\nT72 is ground\nT73 is ground\nT93 is ground\nT94 is ground\nreplacements:X84 -> T93,\nX85 -> T94", fontsize=14, style = filled, fillcolor = violet];
113 -> 337 [arrowhead = none ];
337 -> 134;

338 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
115 -> 338 [arrowhead = none ];
338 -> 116;

339 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
115 -> 339 [arrowhead = none ];
339 -> 117;

340 [label="EVAL with clause\nless(0, s(X94)).\nand substitutionT71 -> 0,\nX94 -> T82,\nT73 -> s(T82)", fontsize=14, style = filled, fillcolor = lightblue];
116 -> 340 [arrowhead = none ];
340 -> 120;

341 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
116 -> 341 [arrowhead = none ];
341 -> 121;

342 [label="EVAL with clause\nless(s(X99), s(X100)) :- less(X99, X100).\nand substitutionX99 -> T87,\nT71 -> s(T87),\nX100 -> T88,\nT73 -> s(T88)", fontsize=14, style = filled, fillcolor = lightblue];
117 -> 342 [arrowhead = none ];
342 -> 123;

343 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
117 -> 343 [arrowhead = none ];
343 -> 124;

344 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
120 -> 344 [arrowhead = none ];
344 -> 122;

345 [label="INSTANCE with matching:\nT71 -> T87\nT73 -> T88", fontsize=14, style = filled, fillcolor = lightgrey];
123 -> 345 [arrowhead = none , style=dashed];
345 -> 111[style=dashed];

346 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
133 -> 346 [arrowhead = none ];
346 -> 137;

347 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
134 -> 347 [arrowhead = none ];
347 -> 235;

348 [label="SPLIT 2\nnew knowledge:\nT71 is ground\nT93 is ground\nT156 is ground\nreplacements:X12 -> T156", fontsize=14, style = filled, fillcolor = violet];
134 -> 348 [arrowhead = none ];
348 -> 236;

349 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
137 -> 349 [arrowhead = none ];
349 -> 138;

350 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
137 -> 350 [arrowhead = none ];
350 -> 139;

351 [label="EVAL with clause\nsplit([], X111, [], []).\nand substitutionT72 -> [],\nT73 -> T101,\nX111 -> T101,\nX84 -> [],\nX85 -> []", fontsize=14, style = filled, fillcolor = lightblue];
138 -> 351 [arrowhead = none ];
351 -> 142;

352 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
138 -> 352 [arrowhead = none ];
352 -> 143;

353 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
139 -> 353 [arrowhead = none ];
353 -> 148;

354 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
139 -> 354 [arrowhead = none ];
354 -> 149;

355 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
142 -> 355 [arrowhead = none ];
355 -> 144;

356 [label="EVAL with clause\nsplit(cons(X140, X141), X142, cons(X140, X143), X144) :- ','(less(X140, X142), split(X141, X142, X143, X144)).\nand substitutionX140 -> T114,\nX141 -> T115,\nT72 -> cons(T114, T115),\nT73 -> T116,\nX142 -> T116,\nX143 -> X145,\nX84 -> cons(T114, X145),\nX85 -> X146,\nX144 -> X146", fontsize=14, style = filled, fillcolor = lightblue];
148 -> 356 [arrowhead = none ];
356 -> 158;

357 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
148 -> 357 [arrowhead = none ];
357 -> 159;

358 [label="EVAL with clause\nsplit(cons(X167, X168), X169, X170, cons(X167, X171)) :- ','(geq(X167, X169), split(X168, X169, X170, X171)).\nand substitutionX167 -> T129,\nX168 -> T130,\nT72 -> cons(T129, T130),\nT73 -> T131,\nX169 -> T131,\nX84 -> X172,\nX170 -> X172,\nX171 -> X173,\nX85 -> cons(T129, X173)", fontsize=14, style = filled, fillcolor = lightblue];
149 -> 358 [arrowhead = none ];
358 -> 201;

359 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
149 -> 359 [arrowhead = none ];
359 -> 202;

360 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
158 -> 360 [arrowhead = none ];
360 -> 163;

361 [label="SPLIT 2\nnew knowledge:\nT114 is ground\nT116 is ground", fontsize=14, style = filled, fillcolor = violet];
158 -> 361 [arrowhead = none ];
361 -> 164;

362 [label="INSTANCE with matching:\nT71 -> T114\nT73 -> T116", fontsize=14, style = filled, fillcolor = lightgrey];
163 -> 362 [arrowhead = none , style=dashed];
362 -> 111[style=dashed];

363 [label="INSTANCE with matching:\nT72 -> T115\nT73 -> T116\nX84 -> X145\nX85 -> X146", fontsize=14, style = filled, fillcolor = lightgrey];
164 -> 363 [arrowhead = none , style=dashed];
363 -> 133[style=dashed];

364 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
201 -> 364 [arrowhead = none ];
364 -> 203;

365 [label="SPLIT 2\nnew knowledge:\nT129 is ground\nT131 is ground", fontsize=14, style = filled, fillcolor = violet];
201 -> 365 [arrowhead = none ];
365 -> 204;

366 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
203 -> 366 [arrowhead = none ];
366 -> 205;

367 [label="INSTANCE with matching:\nT72 -> T130\nT73 -> T131\nX84 -> X172\nX85 -> X173", fontsize=14, style = filled, fillcolor = lightgrey];
204 -> 367 [arrowhead = none , style=dashed];
367 -> 133[style=dashed];

368 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
205 -> 368 [arrowhead = none ];
368 -> 206;

369 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
205 -> 369 [arrowhead = none ];
369 -> 207;

370 [label="EVAL with clause\ngeq(X182, X182).\nand substitutionT129 -> T140,\nX182 -> T140,\nT131 -> T140", fontsize=14, style = filled, fillcolor = lightblue];
206 -> 370 [arrowhead = none ];
370 -> 210;

371 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
206 -> 371 [arrowhead = none ];
371 -> 211;

372 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
207 -> 372 [arrowhead = none ];
372 -> 213;

373 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
207 -> 373 [arrowhead = none ];
373 -> 214;

374 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
210 -> 374 [arrowhead = none ];
374 -> 212;

375 [label="EVAL with clause\ngeq(s(X187), 0).\nand substitutionX187 -> T145,\nT129 -> s(T145),\nT131 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
213 -> 375 [arrowhead = none ];
375 -> 217;

376 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
213 -> 376 [arrowhead = none ];
376 -> 218;

377 [label="EVAL with clause\ngeq(s(X192), s(X193)) :- geq(X192, X193).\nand substitutionX192 -> T150,\nT129 -> s(T150),\nX193 -> T151,\nT131 -> s(T151)", fontsize=14, style = filled, fillcolor = lightblue];
214 -> 377 [arrowhead = none ];
377 -> 221;

378 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
214 -> 378 [arrowhead = none ];
378 -> 223;

379 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
217 -> 379 [arrowhead = none ];
379 -> 219;

380 [label="INSTANCE with matching:\nT129 -> T150\nT131 -> T151", fontsize=14, style = filled, fillcolor = lightgrey];
221 -> 380 [arrowhead = none , style=dashed];
380 -> 203[style=dashed];

381 [label="INSTANCE with matching:\nT1 -> cons(T71, T93)\nT2 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
235 -> 381 [arrowhead = none , style=dashed];
381 -> 3[style=dashed];

382 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
236 -> 382 [arrowhead = none ];
382 -> 237;

383 [label="SPLIT 2\nnew knowledge:\nT94 is ground\nT157 is ground\nreplacements:X13 -> T157", fontsize=14, style = filled, fillcolor = violet];
236 -> 383 [arrowhead = none ];
383 -> 238;

384 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
237 -> 384 [arrowhead = none ];
384 -> 239;

385 [label="INSTANCE with matching:\nT19 -> T156\nT14 -> T73\nT24 -> T157", fontsize=14, style = filled, fillcolor = lightgrey];
238 -> 385 [arrowhead = none , style=dashed];
385 -> 70[style=dashed];

386 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
239 -> 386 [arrowhead = none ];
386 -> 240;

387 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
239 -> 387 [arrowhead = none ];
387 -> 241;

388 [label="EVAL with clause\nqs([], []).\nand substitutionT94 -> [],\nX13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
240 -> 388 [arrowhead = none ];
388 -> 244;

389 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
240 -> 389 [arrowhead = none ];
389 -> 245;

390 [label="EVAL with clause\nqs(cons(X210, X211), X212) :- ','(split(X211, X210, X213, X214), ','(qs(X213, X215), ','(qs(X214, X216), append(X215, cons(X210, X216), X212)))).\nand substitutionX210 -> T162,\nX211 -> T163,\nT94 -> cons(T162, T163),\nX13 -> X217,\nX212 -> X217", fontsize=14, style = filled, fillcolor = lightblue];
241 -> 390 [arrowhead = none ];
390 -> 249;

391 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
241 -> 391 [arrowhead = none ];
391 -> 250;

392 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
244 -> 392 [arrowhead = none ];
392 -> 246;

393 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
249 -> 393 [arrowhead = none ];
393 -> 257;

394 [label="SPLIT 2\nnew knowledge:\nT163 is ground\nT162 is ground\nT166 is ground\nT167 is ground\nreplacements:X213 -> T166,\nX214 -> T167", fontsize=14, style = filled, fillcolor = violet];
249 -> 394 [arrowhead = none ];
394 -> 258;

395 [label="INSTANCE with matching:\nT72 -> T163\nT73 -> T162\nX84 -> X213\nX85 -> X214", fontsize=14, style = filled, fillcolor = lightgrey];
257 -> 395 [arrowhead = none , style=dashed];
395 -> 133[style=dashed];

396 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
258 -> 396 [arrowhead = none ];
396 -> 261;

397 [label="SPLIT 2\nnew knowledge:\nT166 is ground\nT170 is ground\nreplacements:X215 -> T170", fontsize=14, style = filled, fillcolor = violet];
258 -> 397 [arrowhead = none ];
397 -> 262;

398 [label="INSTANCE with matching:\nT94 -> T166\nX13 -> X215", fontsize=14, style = filled, fillcolor = lightgrey];
261 -> 398 [arrowhead = none , style=dashed];
398 -> 237[style=dashed];

399 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
262 -> 399 [arrowhead = none ];
399 -> 265;

400 [label="SPLIT 2\nnew knowledge:\nT167 is ground\nT171 is ground\nreplacements:X216 -> T171", fontsize=14, style = filled, fillcolor = violet];
262 -> 400 [arrowhead = none ];
400 -> 266;

401 [label="INSTANCE with matching:\nT94 -> T167\nX13 -> X216", fontsize=14, style = filled, fillcolor = lightgrey];
265 -> 401 [arrowhead = none , style=dashed];
401 -> 237[style=dashed];

402 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
266 -> 402 [arrowhead = none ];
402 -> 269;

403 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
269 -> 403 [arrowhead = none ];
403 -> 270;

404 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
269 -> 404 [arrowhead = none ];
404 -> 271;

405 [label="EVAL with clause\nappend([], X228, X228).\nand substitutionT170 -> [],\nT162 -> T184,\nT171 -> T185,\nX228 -> cons(T184, T185),\nX217 -> cons(T184, T185)", fontsize=14, style = filled, fillcolor = lightblue];
270 -> 405 [arrowhead = none ];
405 -> 273;

406 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
270 -> 406 [arrowhead = none ];
406 -> 275;

407 [label="EVAL with clause\nappend(cons(X239, X240), X241, cons(X239, X242)) :- append(X240, X241, X242).\nand substitutionX239 -> T194,\nX240 -> T195,\nT170 -> cons(T194, T195),\nT162 -> T196,\nT171 -> T197,\nX241 -> cons(T196, T197),\nX242 -> X243,\nX217 -> cons(T194, X243)", fontsize=14, style = filled, fillcolor = lightblue];
271 -> 407 [arrowhead = none ];
407 -> 278;

408 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
271 -> 408 [arrowhead = none ];
408 -> 279;

409 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
273 -> 409 [arrowhead = none ];
409 -> 276;

410 [label="INSTANCE with matching:\nT170 -> T195\nT162 -> T196\nT171 -> T197\nX217 -> X243", fontsize=14, style = filled, fillcolor = lightgrey];
278 -> 410 [arrowhead = none , style=dashed];
410 -> 266[style=dashed];

411 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
285 -> 411 [arrowhead = none ];
411 -> 287;

412 [label="SPLIT 2\nnew knowledge:\nT212 is ground\nT214 is ground", fontsize=14, style = filled, fillcolor = violet];
285 -> 412 [arrowhead = none ];
412 -> 288;

413 [label="INSTANCE with matching:\nT129 -> T212\nT131 -> T214", fontsize=14, style = filled, fillcolor = lightgrey];
287 -> 413 [arrowhead = none , style=dashed];
413 -> 203[style=dashed];

414 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
288 -> 414 [arrowhead = none ];
414 -> 291;

415 [label="SPLIT 2\nnew knowledge:\nT213 is ground\nT214 is ground\nT221 is ground\nT222 is ground\nreplacements:X267 -> T221,\nX268 -> T222", fontsize=14, style = filled, fillcolor = violet];
288 -> 415 [arrowhead = none ];
415 -> 292;

416 [label="INSTANCE with matching:\nT72 -> T213\nT73 -> T214\nX84 -> X267\nX85 -> X268", fontsize=14, style = filled, fillcolor = lightgrey];
291 -> 416 [arrowhead = none , style=dashed];
416 -> 133[style=dashed];

417 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
292 -> 417 [arrowhead = none ];
417 -> 293;

418 [label="SPLIT 2\nnew knowledge:\nT221 is ground\nT225 is ground\nreplacements:X12 -> T225", fontsize=14, style = filled, fillcolor = violet];
292 -> 418 [arrowhead = none ];
418 -> 294;

419 [label="INSTANCE with matching:\nT94 -> T221\nX13 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
293 -> 419 [arrowhead = none , style=dashed];
419 -> 237[style=dashed];

420 [label="INSTANCE with matching:\nT94 -> cons(T212, T222)\nT156 -> T225\nT73 -> T214", fontsize=14, style = filled, fillcolor = lightgrey];
294 -> 420 [arrowhead = none , style=dashed];
420 -> 236[style=dashed];

}

(2) Obligation:

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

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

(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(cons(T14, []), T9) → U1_GA(T14, T9, pB_in_aaga(X12, X13, T14, T9))
QSA_IN_GA(cons(T14, []), T9) → PB_IN_AAGA(X12, X13, T14, T9)
PB_IN_AAGA(T19, X13, T14, T9) → U11_AAGA(T19, X13, T14, T9, qsE_in_a(T19))
PB_IN_AAGA(T19, X13, T14, T9) → QSE_IN_A(T19)
U11_AAGA(T19, X13, T14, T9, qsE_out_a(T19)) → U12_AAGA(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
U11_AAGA(T19, X13, T14, T9, qsE_out_a(T19)) → PO_IN_AGGA(X13, T19, T14, T9)
PO_IN_AGGA(T24, T19, T14, T9) → U13_AGGA(T24, T19, T14, T9, qsE_in_a(T24))
PO_IN_AGGA(T24, T19, T14, T9) → QSE_IN_A(T24)
U13_AGGA(T24, T19, T14, T9, qsE_out_a(T24)) → U14_AGGA(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
U13_AGGA(T24, T19, T14, T9, qsE_out_a(T24)) → APPENDF_IN_GGGA(T19, T14, T24, T9)
APPENDF_IN_GGGA(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_GGGA(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
APPENDF_IN_GGGA(cons(T49, T50), T51, T52, cons(T49, T54)) → APPENDF_IN_GGGA(T50, T51, T52, T54)
QSA_IN_GA(cons(T73, cons(T71, T72)), T9) → U2_GA(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
QSA_IN_GA(cons(T73, cons(T71, T72)), T9) → PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9)
PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9) → U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9) → LESSG_IN_GG(T71, T73)
LESSG_IN_GG(s(T87), s(T88)) → U5_GG(T87, T88, lessG_in_gg(T87, T88))
LESSG_IN_GG(s(T87), s(T88)) → LESSG_IN_GG(T87, T88)
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, X84, X85, T71, X12, X13, T9)
PP_IN_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9) → U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
PP_IN_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9) → SPLITH_IN_GGAA(T72, T73, T93, T94)
SPLITH_IN_GGAA(cons(T114, T115), T116, cons(T114, X145), X146) → U6_GGAA(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
SPLITH_IN_GGAA(cons(T114, T115), T116, cons(T114, X145), X146) → PI_IN_GGGAA(T114, T116, T115, X145, X146)
PI_IN_GGGAA(T114, T116, T115, X145, X146) → U21_GGGAA(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
PI_IN_GGGAA(T114, T116, T115, X145, X146) → LESSG_IN_GG(T114, T116)
U21_GGGAA(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_GGGAA(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
U21_GGGAA(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → SPLITH_IN_GGAA(T115, T116, X145, X146)
SPLITH_IN_GGAA(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_GGAA(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
SPLITH_IN_GGAA(cons(T129, T130), T131, X172, cons(T129, X173)) → PJ_IN_GGGAA(T129, T131, T130, X172, X173)
PJ_IN_GGGAA(T129, T131, T130, X172, X173) → U23_GGGAA(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
PJ_IN_GGGAA(T129, T131, T130, X172, X173) → GEQK_IN_GG(T129, T131)
GEQK_IN_GG(s(T150), s(T151)) → U8_GG(T150, T151, geqK_in_gg(T150, T151))
GEQK_IN_GG(s(T150), s(T151)) → GEQK_IN_GG(T150, T151)
U23_GGGAA(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_GGGAA(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U23_GGGAA(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → SPLITH_IN_GGAA(T130, T131, X172, X173)
U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → PQ_IN_GGAGAGA(T71, T93, X12, T94, X13, T73, T9)
PQ_IN_GGAGAGA(T71, T93, T156, T94, X13, T73, T9) → U19_GGAGAGA(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
PQ_IN_GGAGAGA(T71, T93, T156, T94, X13, T73, T9) → QSA_IN_GA(cons(T71, T93), T156)
QSA_IN_GA(cons(T214, cons(T212, T213)), T9) → U3_GA(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
QSA_IN_GA(cons(T214, cons(T212, T213)), T9) → PD_IN_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9)
PD_IN_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9) → U33_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
PD_IN_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9) → GEQK_IN_GG(T212, T214)
U33_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
U33_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → PU_IN_GGAAAGAA(T213, T214, X267, X268, X12, T212, X13, T9)
PU_IN_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9) → U35_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
PU_IN_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9) → SPLITH_IN_GGAA(T213, T214, T221, T222)
U35_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
U35_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → PV_IN_GAGGAGA(T221, X12, T212, T222, X13, T214, T9)
PV_IN_GAGGAGA(T221, T225, T212, T222, X13, T214, T9) → U37_GAGGAGA(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
PV_IN_GAGGAGA(T221, T225, T212, T222, X13, T214, T9) → QSL_IN_GA(T221, T225)
QSL_IN_GA(cons(T162, T163), X217) → U9_GA(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
QSL_IN_GA(cons(T162, T163), X217) → PM_IN_GGAAAAA(T163, T162, X213, X214, X215, X216, X217)
PM_IN_GGAAAAA(T163, T162, T166, T167, X215, X216, X217) → U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
PM_IN_GGAAAAA(T163, T162, T166, T167, X215, X216, X217) → SPLITH_IN_GGAA(T163, T162, T166, T167)
U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → PS_IN_GAGAGA(T166, X215, T167, X216, T162, X217)
PS_IN_GAGAGA(T166, T170, T167, X216, T162, X217) → U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
PS_IN_GAGAGA(T166, T170, T167, X216, T162, X217) → QSL_IN_GA(T166, T170)
U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_GAGAGA(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → PT_IN_GAGGA(T167, X216, T170, T162, X217)
PT_IN_GAGGA(T167, T171, T170, T162, X217) → U31_GAGGA(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
PT_IN_GAGGA(T167, T171, T170, T162, X217) → QSL_IN_GA(T167, T171)
U31_GAGGA(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_GAGGA(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
U31_GAGGA(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → APPENDN_IN_GGGA(T170, T162, T171, X217)
APPENDN_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_GGGA(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
APPENDN_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X243)) → APPENDN_IN_GGGA(T195, T196, T197, X243)
U37_GAGGAGA(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_GAGGAGA(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
U37_GAGGAGA(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → PR_IN_GAGGA(cons(T212, T222), X13, T225, T214, T9)
PR_IN_GAGGA(T94, T157, T156, T73, T9) → U25_GAGGA(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
PR_IN_GAGGA(T94, T157, T156, T73, T9) → QSL_IN_GA(T94, T157)
U25_GAGGA(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_GAGGA(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U25_GAGGA(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → APPENDF_IN_GGGA(T156, T73, T157, T9)
U19_GGAGAGA(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_GGAGAGA(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U19_GGAGAGA(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → PR_IN_GAGGA(T94, X13, T156, T73, T9)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

The argument filtering Pi contains the following mapping:
qsA_in_ga(x1, x2) = qsA_in_ga(x1)
[] = []
qsA_out_ga(x1, x2) = qsA_out_ga(x1, x2)
cons(x1, x2) = cons(x1, x2)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
pB_in_aaga(x1, x2, x3, x4) = pB_in_aaga(x3)
U11_aaga(x1, x2, x3, x4, x5) = U11_aaga(x3, x5)
qsE_in_a(x1) = qsE_in_a
qsE_out_a(x1) = qsE_out_a(x1)
U12_aaga(x1, x2, x3, x4, x5) = U12_aaga(x1, x3, x5)
pO_in_agga(x1, x2, x3, x4) = pO_in_agga(x2, x3)
U13_agga(x1, x2, x3, x4, x5) = U13_agga(x2, x3, x5)
U14_agga(x1, x2, x3, x4, x5) = U14_agga(x1, x2, x3, x5)
appendF_in_ggga(x1, x2, x3, x4) = appendF_in_ggga(x1, x2, x3)
appendF_out_ggga(x1, x2, x3, x4) = appendF_out_ggga(x1, x2, x3, x4)
U4_ggga(x1, x2, x3, x4, x5, x6) = U4_ggga(x1, x2, x3, x4, x6)
pO_out_agga(x1, x2, x3, x4) = pO_out_agga(x1, x2, x3, x4)
pB_out_aaga(x1, x2, x3, x4) = pB_out_aaga(x1, x2, x3, x4)
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)
U15_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U15_gggaaaaa(x1, x2, x3, x9)
lessG_in_gg(x1, x2) = lessG_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
lessG_out_gg(x1, x2) = lessG_out_gg(x1, x2)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
U16_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U16_gggaaaaa(x1, x2, x3, x9)
pP_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8) = pP_in_ggaagaaa(x1, x2, x5)
U17_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U17_ggaagaaa(x1, x2, x5, x9)
splitH_in_ggaa(x1, x2, x3, x4) = splitH_in_ggaa(x1, x2)
splitH_out_ggaa(x1, x2, x3, x4) = splitH_out_ggaa(x1, x2, x3, x4)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x2, x3, x6)
pI_in_gggaa(x1, x2, x3, x4, x5) = pI_in_gggaa(x1, x2, x3)
U21_gggaa(x1, x2, x3, x4, x5, x6) = U21_gggaa(x1, x2, x3, x6)
U22_gggaa(x1, x2, x3, x4, x5, x6) = U22_gggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6)
pJ_in_gggaa(x1, x2, x3, x4, x5) = pJ_in_gggaa(x1, x2, x3)
U23_gggaa(x1, x2, x3, x4, x5, x6) = U23_gggaa(x1, x2, x3, x6)
geqK_in_gg(x1, x2) = geqK_in_gg(x1, x2)
geqK_out_gg(x1, x2) = geqK_out_gg(x1, x2)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
U24_gggaa(x1, x2, x3, x4, x5, x6) = U24_gggaa(x1, x2, x3, x6)
pJ_out_gggaa(x1, x2, x3, x4, x5) = pJ_out_gggaa(x1, x2, x3, x4, x5)
pI_out_gggaa(x1, x2, x3, x4, x5) = pI_out_gggaa(x1, x2, x3, x4, x5)
U18_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U18_ggaagaaa(x1, x2, x3, x4, x5, x9)
pQ_in_ggagaga(x1, x2, x3, x4, x5, x6, x7) = pQ_in_ggagaga(x1, x2, x4, x6)
U19_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8) = U19_ggagaga(x1, x2, x4, x6, x8)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x3, x5)
pD_in_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = pD_in_gggaaaaa(x1, x2, x3)
U33_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U33_gggaaaaa(x1, x2, x3, x9)
U34_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U34_gggaaaaa(x1, x2, x3, x9)
pU_in_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = pU_in_ggaaagaa(x1, x2, x6)
U35_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_ggaaagaa(x1, x2, x6, x9)
U36_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U36_ggaaagaa(x1, x2, x3, x4, x6, x9)
pV_in_gaggaga(x1, x2, x3, x4, x5, x6, x7) = pV_in_gaggaga(x1, x3, x4, x6)
U37_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8) = U37_gaggaga(x1, x3, x4, x6, x8)
qsL_in_ga(x1, x2) = qsL_in_ga(x1)
qsL_out_ga(x1, x2) = qsL_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4) = U9_ga(x1, x2, x4)
pM_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = pM_in_ggaaaaa(x1, x2)
U27_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U27_ggaaaaa(x1, x2, x8)
U28_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U28_ggaaaaa(x1, x2, x3, x4, x8)
pS_in_gagaga(x1, x2, x3, x4, x5, x6) = pS_in_gagaga(x1, x3, x5)
U29_gagaga(x1, x2, x3, x4, x5, x6, x7) = U29_gagaga(x1, x3, x5, x7)
U30_gagaga(x1, x2, x3, x4, x5, x6, x7) = U30_gagaga(x1, x2, x3, x5, x7)
pT_in_gagga(x1, x2, x3, x4, x5) = pT_in_gagga(x1, x3, x4)
U31_gagga(x1, x2, x3, x4, x5, x6) = U31_gagga(x1, x3, x4, x6)
U32_gagga(x1, x2, x3, x4, x5, x6) = U32_gagga(x1, x2, x3, x4, x6)
appendN_in_ggga(x1, x2, x3, x4) = appendN_in_ggga(x1, x2, x3)
appendN_out_ggga(x1, x2, x3, x4) = appendN_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4, x5, x6) = U10_ggga(x1, x2, x3, x4, x6)
pT_out_gagga(x1, x2, x3, x4, x5) = pT_out_gagga(x1, x2, x3, x4, x5)
pS_out_gagaga(x1, x2, x3, x4, x5, x6) = pS_out_gagaga(x1, x2, x3, x4, x5, x6)
pM_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = pM_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
U38_gaggaga(x1, x2, x3, x4, x5, x6, x7, x8) = U38_gaggaga(x1, x2, x3, x4, x6, x8)
pR_in_gagga(x1, x2, x3, x4, x5) = pR_in_gagga(x1, x3, x4)
U25_gagga(x1, x2, x3, x4, x5, x6) = U25_gagga(x1, x3, x4, x6)
U26_gagga(x1, x2, x3, x4, x5, x6) = U26_gagga(x1, x2, x3, x4, x6)
pR_out_gagga(x1, x2, x3, x4, x5) = pR_out_gagga(x1, x2, x3, x4, x5)
pV_out_gaggaga(x1, x2, x3, x4, x5, x6, x7) = pV_out_gaggaga(x1, x2, x3, x4, x5, x6, x7)
pU_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8) = pU_out_ggaaagaa(x1, x2, x3, x4, x5, x6, x7, x8)
pD_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = pD_out_gggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)
U20_ggagaga(x1, x2, x3, x4, x5, x6, x7, x8) = U20_ggagaga(x1, x2, x3, x4, x6, x8)
pQ_out_ggagaga(x1, x2, x3, x4, x5, x6, x7) = pQ_out_ggagaga(x1, x2, x3, x4, x5, x6, x7)
pP_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8) = pP_out_ggaagaaa(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_IN_GA(x1, x2) = QSA_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
PB_IN_AAGA(x1, x2, x3, x4) = PB_IN_AAGA(x3)
U11_AAGA(x1, x2, x3, x4, x5) = U11_AAGA(x3, x5)
QSE_IN_A(x1) = QSE_IN_A
U12_AAGA(x1, x2, x3, x4, x5) = U12_AAGA(x1, x3, x5)
PO_IN_AGGA(x1, x2, x3, x4) = PO_IN_AGGA(x2, x3)
U13_AGGA(x1, x2, x3, x4, x5) = U13_AGGA(x2, x3, x5)
U14_AGGA(x1, x2, x3, x4, x5) = U14_AGGA(x1, x2, x3, x5)
APPENDF_IN_GGGA(x1, x2, x3, x4) = APPENDF_IN_GGGA(x1, x2, x3)
U4_GGGA(x1, x2, x3, x4, x5, x6) = U4_GGGA(x1, x2, x3, x4, x6)
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)
U15_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U15_GGGAAAAA(x1, x2, x3, x9)
LESSG_IN_GG(x1, x2) = LESSG_IN_GG(x1, x2)
U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3)
U16_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U16_GGGAAAAA(x1, x2, x3, x9)
PP_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8) = PP_IN_GGAAGAAA(x1, x2, x5)
U17_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U17_GGAAGAAA(x1, x2, x5, x9)
SPLITH_IN_GGAA(x1, x2, x3, x4) = SPLITH_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x1, x2, x3, x6)
PI_IN_GGGAA(x1, x2, x3, x4, x5) = PI_IN_GGGAA(x1, x2, x3)
U21_GGGAA(x1, x2, x3, x4, x5, x6) = U21_GGGAA(x1, x2, x3, x6)
U22_GGGAA(x1, x2, x3, x4, x5, x6) = U22_GGGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6)
PJ_IN_GGGAA(x1, x2, x3, x4, x5) = PJ_IN_GGGAA(x1, x2, x3)
U23_GGGAA(x1, x2, x3, x4, x5, x6) = U23_GGGAA(x1, x2, x3, x6)
GEQK_IN_GG(x1, x2) = GEQK_IN_GG(x1, x2)
U8_GG(x1, x2, x3) = U8_GG(x1, x2, x3)
U24_GGGAA(x1, x2, x3, x4, x5, x6) = U24_GGGAA(x1, x2, x3, x6)
U18_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U18_GGAAGAAA(x1, x2, x3, x4, x5, x9)
PQ_IN_GGAGAGA(x1, x2, x3, x4, x5, x6, x7) = PQ_IN_GGAGAGA(x1, x2, x4, x6)
U19_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U19_GGAGAGA(x1, x2, x4, x6, x8)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x3, x5)
PD_IN_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = PD_IN_GGGAAAAA(x1, x2, x3)
U33_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U33_GGGAAAAA(x1, x2, x3, x9)
U34_GGGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U34_GGGAAAAA(x1, x2, x3, x9)
PU_IN_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8) = PU_IN_GGAAAGAA(x1, x2, x6)
U35_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_GGAAAGAA(x1, x2, x6, x9)
U36_GGAAAGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U36_GGAAAGAA(x1, x2, x3, x4, x6, x9)
PV_IN_GAGGAGA(x1, x2, x3, x4, x5, x6, x7) = PV_IN_GAGGAGA(x1, x3, x4, x6)
U37_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U37_GAGGAGA(x1, x3, x4, x6, x8)
QSL_IN_GA(x1, x2) = QSL_IN_GA(x1)
U9_GA(x1, x2, x3, x4) = U9_GA(x1, x2, x4)
PM_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PM_IN_GGAAAAA(x1, x2)
U27_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U27_GGAAAAA(x1, x2, x8)
U28_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U28_GGAAAAA(x1, x2, x3, x4, x8)
PS_IN_GAGAGA(x1, x2, x3, x4, x5, x6) = PS_IN_GAGAGA(x1, x3, x5)
U29_GAGAGA(x1, x2, x3, x4, x5, x6, x7) = U29_GAGAGA(x1, x3, x5, x7)
U30_GAGAGA(x1, x2, x3, x4, x5, x6, x7) = U30_GAGAGA(x1, x2, x3, x5, x7)
PT_IN_GAGGA(x1, x2, x3, x4, x5) = PT_IN_GAGGA(x1, x3, x4)
U31_GAGGA(x1, x2, x3, x4, x5, x6) = U31_GAGGA(x1, x3, x4, x6)
U32_GAGGA(x1, x2, x3, x4, x5, x6) = U32_GAGGA(x1, x2, x3, x4, x6)
APPENDN_IN_GGGA(x1, x2, x3, x4) = APPENDN_IN_GGGA(x1, x2, x3)
U10_GGGA(x1, x2, x3, x4, x5, x6) = U10_GGGA(x1, x2, x3, x4, x6)
U38_GAGGAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U38_GAGGAGA(x1, x2, x3, x4, x6, x8)
PR_IN_GAGGA(x1, x2, x3, x4, x5) = PR_IN_GAGGA(x1, x3, x4)
U25_GAGGA(x1, x2, x3, x4, x5, x6) = U25_GAGGA(x1, x3, x4, x6)
U26_GAGGA(x1, x2, x3, x4, x5, x6) = U26_GAGGA(x1, x2, x3, x4, x6)
U20_GGAGAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U20_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(cons(T14, []), T9) → U1_GA(T14, T9, pB_in_aaga(X12, X13, T14, T9))
QSA_IN_GA(cons(T14, []), T9) → PB_IN_AAGA(X12, X13, T14, T9)
PB_IN_AAGA(T19, X13, T14, T9) → U11_AAGA(T19, X13, T14, T9, qsE_in_a(T19))
PB_IN_AAGA(T19, X13, T14, T9) → QSE_IN_A(T19)
U11_AAGA(T19, X13, T14, T9, qsE_out_a(T19)) → U12_AAGA(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
U11_AAGA(T19, X13, T14, T9, qsE_out_a(T19)) → PO_IN_AGGA(X13, T19, T14, T9)
PO_IN_AGGA(T24, T19, T14, T9) → U13_AGGA(T24, T19, T14, T9, qsE_in_a(T24))
PO_IN_AGGA(T24, T19, T14, T9) → QSE_IN_A(T24)
U13_AGGA(T24, T19, T14, T9, qsE_out_a(T24)) → U14_AGGA(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
U13_AGGA(T24, T19, T14, T9, qsE_out_a(T24)) → APPENDF_IN_GGGA(T19, T14, T24, T9)
APPENDF_IN_GGGA(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_GGGA(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
APPENDF_IN_GGGA(cons(T49, T50), T51, T52, cons(T49, T54)) → APPENDF_IN_GGGA(T50, T51, T52, T54)
QSA_IN_GA(cons(T73, cons(T71, T72)), T9) → U2_GA(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
QSA_IN_GA(cons(T73, cons(T71, T72)), T9) → PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9)
PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9) → U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9) → LESSG_IN_GG(T71, T73)
LESSG_IN_GG(s(T87), s(T88)) → U5_GG(T87, T88, lessG_in_gg(T87, T88))
LESSG_IN_GG(s(T87), s(T88)) → LESSG_IN_GG(T87, T88)
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, X84, X85, T71, X12, X13, T9)
PP_IN_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9) → U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
PP_IN_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9) → SPLITH_IN_GGAA(T72, T73, T93, T94)
SPLITH_IN_GGAA(cons(T114, T115), T116, cons(T114, X145), X146) → U6_GGAA(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
SPLITH_IN_GGAA(cons(T114, T115), T116, cons(T114, X145), X146) → PI_IN_GGGAA(T114, T116, T115, X145, X146)
PI_IN_GGGAA(T114, T116, T115, X145, X146) → U21_GGGAA(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
PI_IN_GGGAA(T114, T116, T115, X145, X146) → LESSG_IN_GG(T114, T116)
U21_GGGAA(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_GGGAA(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
U21_GGGAA(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → SPLITH_IN_GGAA(T115, T116, X145, X146)
SPLITH_IN_GGAA(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_GGAA(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
SPLITH_IN_GGAA(cons(T129, T130), T131, X172, cons(T129, X173)) → PJ_IN_GGGAA(T129, T131, T130, X172, X173)
PJ_IN_GGGAA(T129, T131, T130, X172, X173) → U23_GGGAA(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
PJ_IN_GGGAA(T129, T131, T130, X172, X173) → GEQK_IN_GG(T129, T131)
GEQK_IN_GG(s(T150), s(T151)) → U8_GG(T150, T151, geqK_in_gg(T150, T151))
GEQK_IN_GG(s(T150), s(T151)) → GEQK_IN_GG(T150, T151)
U23_GGGAA(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_GGGAA(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U23_GGGAA(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → SPLITH_IN_GGAA(T130, T131, X172, X173)
U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → PQ_IN_GGAGAGA(T71, T93, X12, T94, X13, T73, T9)
PQ_IN_GGAGAGA(T71, T93, T156, T94, X13, T73, T9) → U19_GGAGAGA(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
PQ_IN_GGAGAGA(T71, T93, T156, T94, X13, T73, T9) → QSA_IN_GA(cons(T71, T93), T156)
QSA_IN_GA(cons(T214, cons(T212, T213)), T9) → U3_GA(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
QSA_IN_GA(cons(T214, cons(T212, T213)), T9) → PD_IN_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9)
PD_IN_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9) → U33_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
PD_IN_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9) → GEQK_IN_GG(T212, T214)
U33_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
U33_GGGAAAAA(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → PU_IN_GGAAAGAA(T213, T214, X267, X268, X12, T212, X13, T9)
PU_IN_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9) → U35_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
PU_IN_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9) → SPLITH_IN_GGAA(T213, T214, T221, T222)
U35_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
U35_GGAAAGAA(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → PV_IN_GAGGAGA(T221, X12, T212, T222, X13, T214, T9)
PV_IN_GAGGAGA(T221, T225, T212, T222, X13, T214, T9) → U37_GAGGAGA(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
PV_IN_GAGGAGA(T221, T225, T212, T222, X13, T214, T9) → QSL_IN_GA(T221, T225)
QSL_IN_GA(cons(T162, T163), X217) → U9_GA(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
QSL_IN_GA(cons(T162, T163), X217) → PM_IN_GGAAAAA(T163, T162, X213, X214, X215, X216, X217)
PM_IN_GGAAAAA(T163, T162, T166, T167, X215, X216, X217) → U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
PM_IN_GGAAAAA(T163, T162, T166, T167, X215, X216, X217) → SPLITH_IN_GGAA(T163, T162, T166, T167)
U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → PS_IN_GAGAGA(T166, X215, T167, X216, T162, X217)
PS_IN_GAGAGA(T166, T170, T167, X216, T162, X217) → U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
PS_IN_GAGAGA(T166, T170, T167, X216, T162, X217) → QSL_IN_GA(T166, T170)
U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_GAGAGA(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → PT_IN_GAGGA(T167, X216, T170, T162, X217)
PT_IN_GAGGA(T167, T171, T170, T162, X217) → U31_GAGGA(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
PT_IN_GAGGA(T167, T171, T170, T162, X217) → QSL_IN_GA(T167, T171)
U31_GAGGA(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_GAGGA(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
U31_GAGGA(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → APPENDN_IN_GGGA(T170, T162, T171, X217)
APPENDN_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_GGGA(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
APPENDN_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X243)) → APPENDN_IN_GGGA(T195, T196, T197, X243)
U37_GAGGAGA(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_GAGGAGA(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
U37_GAGGAGA(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → PR_IN_GAGGA(cons(T212, T222), X13, T225, T214, T9)
PR_IN_GAGGA(T94, T157, T156, T73, T9) → U25_GAGGA(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
PR_IN_GAGGA(T94, T157, T156, T73, T9) → QSL_IN_GA(T94, T157)
U25_GAGGA(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_GAGGA(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U25_GAGGA(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → APPENDF_IN_GGGA(T156, T73, T157, T9)
U19_GGAGAGA(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_GGAGAGA(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U19_GGAGAGA(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → PR_IN_GAGGA(T94, X13, T156, T73, T9)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

(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:

APPENDN_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X243)) → APPENDN_IN_GGGA(T195, T196, T197, X243)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

(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:

APPENDN_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X243)) → APPENDN_IN_GGGA(T195, T196, T197, X243)

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

APPENDN_IN_GGGA(cons(T194, T195), T196, T197) → APPENDN_IN_GGGA(T195, T196, T197)

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:

APPENDN_IN_GGGA(cons(T194, T195), T196, T197) → APPENDN_IN_GGGA(T195, T196, T197)
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:

GEQK_IN_GG(s(T150), s(T151)) → GEQK_IN_GG(T150, T151)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

(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:

GEQK_IN_GG(s(T150), s(T151)) → GEQK_IN_GG(T150, T151)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

GEQK_IN_GG(s(T150), s(T151)) → GEQK_IN_GG(T150, T151)

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:

GEQK_IN_GG(s(T150), s(T151)) → GEQK_IN_GG(T150, T151)
The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

LESSG_IN_GG(s(T87), s(T88)) → LESSG_IN_GG(T87, T88)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

(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:

LESSG_IN_GG(s(T87), s(T88)) → LESSG_IN_GG(T87, T88)

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:

LESSG_IN_GG(s(T87), s(T88)) → LESSG_IN_GG(T87, T88)

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:

LESSG_IN_GG(s(T87), s(T88)) → LESSG_IN_GG(T87, T88)
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:

PI_IN_GGGAA(T114, T116, T115, X145, X146) → U21_GGGAA(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_GGGAA(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → SPLITH_IN_GGAA(T115, T116, X145, X146)
SPLITH_IN_GGAA(cons(T114, T115), T116, cons(T114, X145), X146) → PI_IN_GGGAA(T114, T116, T115, X145, X146)
SPLITH_IN_GGAA(cons(T129, T130), T131, X172, cons(T129, X173)) → PJ_IN_GGGAA(T129, T131, T130, X172, X173)
PJ_IN_GGGAA(T129, T131, T130, X172, X173) → U23_GGGAA(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
U23_GGGAA(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → SPLITH_IN_GGAA(T130, T131, X172, X173)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

(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:

PI_IN_GGGAA(T114, T116, T115, X145, X146) → U21_GGGAA(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_GGGAA(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → SPLITH_IN_GGAA(T115, T116, X145, X146)
SPLITH_IN_GGAA(cons(T114, T115), T116, cons(T114, X145), X146) → PI_IN_GGGAA(T114, T116, T115, X145, X146)
SPLITH_IN_GGAA(cons(T129, T130), T131, X172, cons(T129, X173)) → PJ_IN_GGGAA(T129, T131, T130, X172, X173)
PJ_IN_GGGAA(T129, T131, T130, X172, X173) → U23_GGGAA(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
U23_GGGAA(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → SPLITH_IN_GGAA(T130, T131, X172, X173)

The TRS R consists of the following rules:

lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))

The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x1, x2)
lessG_in_gg(x1, x2) = lessG_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
lessG_out_gg(x1, x2) = lessG_out_gg(x1, x2)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
geqK_in_gg(x1, x2) = geqK_in_gg(x1, x2)
geqK_out_gg(x1, x2) = geqK_out_gg(x1, x2)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
SPLITH_IN_GGAA(x1, x2, x3, x4) = SPLITH_IN_GGAA(x1, x2)
PI_IN_GGGAA(x1, x2, x3, x4, x5) = PI_IN_GGGAA(x1, x2, x3)
U21_GGGAA(x1, x2, x3, x4, x5, x6) = U21_GGGAA(x1, x2, x3, x6)
PJ_IN_GGGAA(x1, x2, x3, x4, x5) = PJ_IN_GGGAA(x1, x2, x3)
U23_GGGAA(x1, x2, x3, x4, x5, x6) = U23_GGGAA(x1, x2, x3, x6)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

PI_IN_GGGAA(T114, T116, T115) → U21_GGGAA(T114, T116, T115, lessG_in_gg(T114, T116))
U21_GGGAA(T114, T116, T115, lessG_out_gg(T114, T116)) → SPLITH_IN_GGAA(T115, T116)
SPLITH_IN_GGAA(cons(T114, T115), T116) → PI_IN_GGGAA(T114, T116, T115)
SPLITH_IN_GGAA(cons(T129, T130), T131) → PJ_IN_GGGAA(T129, T131, T130)
PJ_IN_GGGAA(T129, T131, T130) → U23_GGGAA(T129, T131, T130, geqK_in_gg(T129, T131))
U23_GGGAA(T129, T131, T130, geqK_out_gg(T129, T131)) → SPLITH_IN_GGAA(T130, T131)

The TRS R consists of the following rules:

lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))

The set Q consists of the following terms:

lessG_in_gg(x0, x1)
geqK_in_gg(x0, x1)
U5_gg(x0, x1, x2)
U8_gg(x0, x1, x2)

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

(33) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

U21_GGGAA(T114, T116, T115, lessG_out_gg(T114, T116)) → SPLITH_IN_GGAA(T115, T116)
The graph contains the following edges 3 >= 1, 2 >= 2, 4 > 2
SPLITH_IN_GGAA(cons(T114, T115), T116) → PI_IN_GGGAA(T114, T116, T115)
The graph contains the following edges 1 > 1, 2 >= 2, 1 > 3
SPLITH_IN_GGAA(cons(T129, T130), T131) → PJ_IN_GGGAA(T129, T131, T130)
The graph contains the following edges 1 > 1, 2 >= 2, 1 > 3
PI_IN_GGGAA(T114, T116, T115) → U21_GGGAA(T114, T116, T115, lessG_in_gg(T114, T116))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
U23_GGGAA(T129, T131, T130, geqK_out_gg(T129, T131)) → SPLITH_IN_GGAA(T130, T131)
The graph contains the following edges 3 >= 1, 2 >= 2, 4 > 2
PJ_IN_GGGAA(T129, T131, T130) → U23_GGGAA(T129, T131, T130, geqK_in_gg(T129, T131))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3

(34) YES

(35) Obligation:

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

PM_IN_GGAAAAA(T163, T162, T166, T167, X215, X216, X217) → U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → PS_IN_GAGAGA(T166, X215, T167, X216, T162, X217)
PS_IN_GAGAGA(T166, T170, T167, X216, T162, X217) → U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → PT_IN_GAGGA(T167, X216, T170, T162, X217)
PT_IN_GAGGA(T167, T171, T170, T162, X217) → QSL_IN_GA(T167, T171)
QSL_IN_GA(cons(T162, T163), X217) → PM_IN_GGAAAAA(T163, T162, X213, X214, X215, X216, X217)
PS_IN_GAGAGA(T166, T170, T167, X216, T162, X217) → QSL_IN_GA(T166, T170)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

(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:

PM_IN_GGAAAAA(T163, T162, T166, T167, X215, X216, X217) → U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_GGAAAAA(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → PS_IN_GAGAGA(T166, X215, T167, X216, T162, X217)
PS_IN_GAGAGA(T166, T170, T167, X216, T162, X217) → U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_GAGAGA(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → PT_IN_GAGGA(T167, X216, T170, T162, X217)
PT_IN_GAGGA(T167, T171, T170, T162, X217) → QSL_IN_GA(T167, T171)
QSL_IN_GA(cons(T162, T163), X217) → PM_IN_GGAAAAA(T163, T162, X213, X214, X215, X216, X217)
PS_IN_GAGAGA(T166, T170, T167, X216, T162, X217) → QSL_IN_GA(T166, T170)

The TRS R consists of the following rules:

splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))

The argument filtering Pi contains the following mapping:
[] = []
cons(x1, x2) = cons(x1, x2)
lessG_in_gg(x1, x2) = lessG_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
lessG_out_gg(x1, x2) = lessG_out_gg(x1, x2)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
splitH_in_ggaa(x1, x2, x3, x4) = splitH_in_ggaa(x1, x2)
splitH_out_ggaa(x1, x2, x3, x4) = splitH_out_ggaa(x1, x2, x3, x4)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x1, x2, x3, x6)
pI_in_gggaa(x1, x2, x3, x4, x5) = pI_in_gggaa(x1, x2, x3)
U21_gggaa(x1, x2, x3, x4, x5, x6) = U21_gggaa(x1, x2, x3, x6)
U22_gggaa(x1, x2, x3, x4, x5, x6) = U22_gggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6)
pJ_in_gggaa(x1, x2, x3, x4, x5) = pJ_in_gggaa(x1, x2, x3)
U23_gggaa(x1, x2, x3, x4, x5, x6) = U23_gggaa(x1, x2, x3, x6)
geqK_in_gg(x1, x2) = geqK_in_gg(x1, x2)
geqK_out_gg(x1, x2) = geqK_out_gg(x1, x2)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
U24_gggaa(x1, x2, x3, x4, x5, x6) = U24_gggaa(x1, x2, x3, x6)
pJ_out_gggaa(x1, x2, x3, x4, x5) = pJ_out_gggaa(x1, x2, x3, x4, x5)
pI_out_gggaa(x1, x2, x3, x4, x5) = pI_out_gggaa(x1, x2, x3, x4, x5)
qsL_in_ga(x1, x2) = qsL_in_ga(x1)
qsL_out_ga(x1, x2) = qsL_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4) = U9_ga(x1, x2, x4)
pM_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = pM_in_ggaaaaa(x1, x2)
U27_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U27_ggaaaaa(x1, x2, x8)
U28_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U28_ggaaaaa(x1, x2, x3, x4, x8)
pS_in_gagaga(x1, x2, x3, x4, x5, x6) = pS_in_gagaga(x1, x3, x5)
U29_gagaga(x1, x2, x3, x4, x5, x6, x7) = U29_gagaga(x1, x3, x5, x7)
U30_gagaga(x1, x2, x3, x4, x5, x6, x7) = U30_gagaga(x1, x2, x3, x5, x7)
pT_in_gagga(x1, x2, x3, x4, x5) = pT_in_gagga(x1, x3, x4)
U31_gagga(x1, x2, x3, x4, x5, x6) = U31_gagga(x1, x3, x4, x6)
U32_gagga(x1, x2, x3, x4, x5, x6) = U32_gagga(x1, x2, x3, x4, x6)
appendN_in_ggga(x1, x2, x3, x4) = appendN_in_ggga(x1, x2, x3)
appendN_out_ggga(x1, x2, x3, x4) = appendN_out_ggga(x1, x2, x3, x4)
U10_ggga(x1, x2, x3, x4, x5, x6) = U10_ggga(x1, x2, x3, x4, x6)
pT_out_gagga(x1, x2, x3, x4, x5) = pT_out_gagga(x1, x2, x3, x4, x5)
pS_out_gagaga(x1, x2, x3, x4, x5, x6) = pS_out_gagaga(x1, x2, x3, x4, x5, x6)
pM_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7) = pM_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)
QSL_IN_GA(x1, x2) = QSL_IN_GA(x1)
PM_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7) = PM_IN_GGAAAAA(x1, x2)
U27_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U27_GGAAAAA(x1, x2, x8)
PS_IN_GAGAGA(x1, x2, x3, x4, x5, x6) = PS_IN_GAGAGA(x1, x3, x5)
U29_GAGAGA(x1, x2, x3, x4, x5, x6, x7) = U29_GAGAGA(x1, x3, x5, x7)
PT_IN_GAGGA(x1, x2, x3, x4, x5) = PT_IN_GAGGA(x1, x3, x4)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

PM_IN_GGAAAAA(T163, T162) → U27_GGAAAAA(T163, T162, splitH_in_ggaa(T163, T162))
U27_GGAAAAA(T163, T162, splitH_out_ggaa(T163, T162, T166, T167)) → PS_IN_GAGAGA(T166, T167, T162)
PS_IN_GAGAGA(T166, T167, T162) → U29_GAGAGA(T166, T167, T162, qsL_in_ga(T166))
U29_GAGAGA(T166, T167, T162, qsL_out_ga(T166, T170)) → PT_IN_GAGGA(T167, T170, T162)
PT_IN_GAGGA(T167, T170, T162) → QSL_IN_GA(T167)
QSL_IN_GA(cons(T162, T163)) → PM_IN_GGAAAAA(T163, T162)
PS_IN_GAGAGA(T166, T167, T162) → QSL_IN_GA(T166)

The TRS R consists of the following rules:

splitH_in_ggaa([], T101) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116) → U6_ggaa(T114, T115, T116, pI_in_gggaa(T114, T116, T115))
splitH_in_ggaa(cons(T129, T130), T131) → U7_ggaa(T129, T130, T131, pJ_in_gggaa(T129, T131, T130))
qsL_in_ga([]) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163)) → U9_ga(T162, T163, pM_in_ggaaaaa(T163, T162))
U6_ggaa(T114, T115, T116, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U7_ggaa(T129, T130, T131, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U9_ga(T162, T163, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
pI_in_gggaa(T114, T116, T115) → U21_gggaa(T114, T116, T115, lessG_in_gg(T114, T116))
pJ_in_gggaa(T129, T131, T130) → U23_gggaa(T129, T131, T130, geqK_in_gg(T129, T131))
pM_in_ggaaaaa(T163, T162) → U27_ggaaaaa(T163, T162, splitH_in_ggaa(T163, T162))
U21_gggaa(T114, T116, T115, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, splitH_in_ggaa(T115, T116))
U23_gggaa(T129, T131, T130, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, splitH_in_ggaa(T130, T131))
U27_ggaaaaa(T163, T162, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, pS_in_gagaga(T166, T167, T162))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U22_gggaa(T114, T116, T115, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U24_gggaa(T129, T131, T130, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U28_ggaaaaa(T163, T162, T166, T167, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
pS_in_gagaga(T166, T167, T162) → U29_gagaga(T166, T167, T162, qsL_in_ga(T166))
U29_gagaga(T166, T167, T162, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, T162, pT_in_gagga(T167, T170, T162))
U30_gagaga(T166, T170, T167, T162, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
pT_in_gagga(T167, T170, T162) → U31_gagga(T167, T170, T162, qsL_in_ga(T167))
U31_gagga(T167, T170, T162, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, appendN_in_ggga(T170, T162, T171))
U32_gagga(T167, T171, T170, T162, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
appendN_in_ggga([], T184, T185) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197) → U10_ggga(T194, T195, T196, T197, appendN_in_ggga(T195, T196, T197))
U10_ggga(T194, T195, T196, T197, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))

The set Q consists of the following terms:

splitH_in_ggaa(x0, x1)
qsL_in_ga(x0)
U6_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U9_ga(x0, x1, x2)
pI_in_gggaa(x0, x1, x2)
pJ_in_gggaa(x0, x1, x2)
pM_in_ggaaaaa(x0, x1)
U21_gggaa(x0, x1, x2, x3)
U23_gggaa(x0, x1, x2, x3)
U27_ggaaaaa(x0, x1, x2)
lessG_in_gg(x0, x1)
U22_gggaa(x0, x1, x2, x3)
geqK_in_gg(x0, x1)
U24_gggaa(x0, x1, x2, x3)
U28_ggaaaaa(x0, x1, x2, x3, x4)
U5_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
pS_in_gagaga(x0, x1, x2)
U29_gagaga(x0, x1, x2, x3)
U30_gagaga(x0, x1, x2, x3, x4)
pT_in_gagga(x0, x1, x2)
U31_gagga(x0, x1, x2, x3)
U32_gagga(x0, x1, x2, x3, x4)
appendN_in_ggga(x0, x1, x2)
U10_ggga(x0, x1, x2, x3, x4)

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

(40) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PM_IN_GGAAAAA(T163, T162) → U27_GGAAAAA(T163, T162, splitH_in_ggaa(T163, T162))

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

POL(0) = 0
POL(PM_IN_GGAAAAA(x₁, x₂)) = 1 + x₁
POL(PS_IN_GAGAGA(x₁, x₂, x₃)) = x₁ + x₂
POL(PT_IN_GAGGA(x₁, x₂, x₃)) = x₁
POL(QSL_IN_GA(x₁)) = x₁
POL(U10_ggga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U21_gggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U22_gggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U23_gggaa(x₁, x₂, x₃, x₄)) = x₃
POL(U24_gggaa(x₁, x₂, x₃, x₄)) = x₄
POL(U27_GGAAAAA(x₁, x₂, x₃)) = x₃
POL(U27_ggaaaaa(x₁, x₂, x₃)) = 0
POL(U28_ggaaaaa(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U29_GAGAGA(x₁, x₂, x₃, x₄)) = x₂
POL(U29_gagaga(x₁, x₂, x₃, x₄)) = 0
POL(U30_gagaga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U31_gagga(x₁, x₂, x₃, x₄)) = 0
POL(U32_gagga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U5_gg(x₁, x₂, x₃)) = 0
POL(U6_ggaa(x₁, x₂, x₃, x₄)) = x₄
POL(U7_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U8_gg(x₁, x₂, x₃)) = 0
POL(U9_ga(x₁, x₂, x₃)) = 0
POL([]) = 0
POL(appendN_in_ggga(x₁, x₂, x₃)) = 0
POL(appendN_out_ggga(x₁, x₂, x₃, x₄)) = 0
POL(cons(x₁, x₂)) = 1 + x₂
POL(geqK_in_gg(x₁, x₂)) = 0
POL(geqK_out_gg(x₁, x₂)) = 0
POL(lessG_in_gg(x₁, x₂)) = 0
POL(lessG_out_gg(x₁, x₂)) = 0
POL(pI_in_gggaa(x₁, x₂, x₃)) = 1 + x₃
POL(pI_out_gggaa(x₁, x₂, x₃, x₄, x₅)) = 1 + x₄ + x₅
POL(pJ_in_gggaa(x₁, x₂, x₃)) = x₃
POL(pJ_out_gggaa(x₁, x₂, x₃, x₄, x₅)) = x₄ + x₅
POL(pM_in_ggaaaaa(x₁, x₂)) = 0
POL(pM_out_ggaaaaa(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 0
POL(pS_in_gagaga(x₁, x₂, x₃)) = 0
POL(pS_out_gagaga(x₁, x₂, x₃, x₄, x₅, x₆)) = 0
POL(pT_in_gagga(x₁, x₂, x₃)) = 0
POL(pT_out_gagga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(qsL_in_ga(x₁)) = 0
POL(qsL_out_ga(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(splitH_in_ggaa(x₁, x₂)) = x₁
POL(splitH_out_ggaa(x₁, x₂, x₃, x₄)) = x₃ + x₄

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

splitH_in_ggaa([], T101) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116) → U6_ggaa(T114, T115, T116, pI_in_gggaa(T114, T116, T115))
splitH_in_ggaa(cons(T129, T130), T131) → U7_ggaa(T129, T130, T131, pJ_in_gggaa(T129, T131, T130))
U6_ggaa(T114, T115, T116, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
pI_in_gggaa(T114, T116, T115) → U21_gggaa(T114, T116, T115, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, splitH_in_ggaa(T115, T116))
U22_gggaa(T114, T116, T115, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U7_ggaa(T129, T130, T131, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
pJ_in_gggaa(T129, T131, T130) → U23_gggaa(T129, T131, T130, geqK_in_gg(T129, T131))
U23_gggaa(T129, T131, T130, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, splitH_in_ggaa(T130, T131))
U24_gggaa(T129, T131, T130, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)

(41) Obligation:

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

U27_GGAAAAA(T163, T162, splitH_out_ggaa(T163, T162, T166, T167)) → PS_IN_GAGAGA(T166, T167, T162)
PS_IN_GAGAGA(T166, T167, T162) → U29_GAGAGA(T166, T167, T162, qsL_in_ga(T166))
U29_GAGAGA(T166, T167, T162, qsL_out_ga(T166, T170)) → PT_IN_GAGGA(T167, T170, T162)
PT_IN_GAGGA(T167, T170, T162) → QSL_IN_GA(T167)
QSL_IN_GA(cons(T162, T163)) → PM_IN_GGAAAAA(T163, T162)
PS_IN_GAGAGA(T166, T167, T162) → QSL_IN_GA(T166)

The TRS R consists of the following rules:

splitH_in_ggaa([], T101) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116) → U6_ggaa(T114, T115, T116, pI_in_gggaa(T114, T116, T115))
splitH_in_ggaa(cons(T129, T130), T131) → U7_ggaa(T129, T130, T131, pJ_in_gggaa(T129, T131, T130))
qsL_in_ga([]) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163)) → U9_ga(T162, T163, pM_in_ggaaaaa(T163, T162))
U6_ggaa(T114, T115, T116, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U7_ggaa(T129, T130, T131, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U9_ga(T162, T163, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
pI_in_gggaa(T114, T116, T115) → U21_gggaa(T114, T116, T115, lessG_in_gg(T114, T116))
pJ_in_gggaa(T129, T131, T130) → U23_gggaa(T129, T131, T130, geqK_in_gg(T129, T131))
pM_in_ggaaaaa(T163, T162) → U27_ggaaaaa(T163, T162, splitH_in_ggaa(T163, T162))
U21_gggaa(T114, T116, T115, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, splitH_in_ggaa(T115, T116))
U23_gggaa(T129, T131, T130, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, splitH_in_ggaa(T130, T131))
U27_ggaaaaa(T163, T162, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, pS_in_gagaga(T166, T167, T162))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U22_gggaa(T114, T116, T115, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U24_gggaa(T129, T131, T130, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U28_ggaaaaa(T163, T162, T166, T167, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
pS_in_gagaga(T166, T167, T162) → U29_gagaga(T166, T167, T162, qsL_in_ga(T166))
U29_gagaga(T166, T167, T162, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, T162, pT_in_gagga(T167, T170, T162))
U30_gagaga(T166, T170, T167, T162, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
pT_in_gagga(T167, T170, T162) → U31_gagga(T167, T170, T162, qsL_in_ga(T167))
U31_gagga(T167, T170, T162, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, appendN_in_ggga(T170, T162, T171))
U32_gagga(T167, T171, T170, T162, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
appendN_in_ggga([], T184, T185) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197) → U10_ggga(T194, T195, T196, T197, appendN_in_ggga(T195, T196, T197))
U10_ggga(T194, T195, T196, T197, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))

The set Q consists of the following terms:

splitH_in_ggaa(x0, x1)
qsL_in_ga(x0)
U6_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U9_ga(x0, x1, x2)
pI_in_gggaa(x0, x1, x2)
pJ_in_gggaa(x0, x1, x2)
pM_in_ggaaaaa(x0, x1)
U21_gggaa(x0, x1, x2, x3)
U23_gggaa(x0, x1, x2, x3)
U27_ggaaaaa(x0, x1, x2)
lessG_in_gg(x0, x1)
U22_gggaa(x0, x1, x2, x3)
geqK_in_gg(x0, x1)
U24_gggaa(x0, x1, x2, x3)
U28_ggaaaaa(x0, x1, x2, x3, x4)
U5_gg(x0, x1, x2)
U8_gg(x0, x1, x2)
pS_in_gagaga(x0, x1, x2)
U29_gagaga(x0, x1, x2, x3)
U30_gagaga(x0, x1, x2, x3, x4)
pT_in_gagga(x0, x1, x2)
U31_gagga(x0, x1, x2, x3)
U32_gagga(x0, x1, x2, x3, x4)
appendN_in_ggga(x0, x1, x2)
U10_ggga(x0, x1, x2, x3, x4)

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

(42) DependencyGraphProof (EQUIVALENT transformation)

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

(43) TRUE

(44) Obligation:

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

APPENDF_IN_GGGA(cons(T49, T50), T51, T52, cons(T49, T54)) → APPENDF_IN_GGGA(T50, T51, T52, T54)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

APPENDF_IN_GGGA(cons(T49, T50), T51, T52, cons(T49, T54)) → APPENDF_IN_GGGA(T50, T51, T52, T54)

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

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

APPENDF_IN_GGGA(cons(T49, T50), T51, T52) → APPENDF_IN_GGGA(T50, T51, T52)

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

(49) 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:

APPENDF_IN_GGGA(cons(T49, T50), T51, T52) → APPENDF_IN_GGGA(T50, T51, T52)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(50) YES

(51) Obligation:

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

QSA_IN_GA(cons(T73, cons(T71, T72)), T9) → PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9)
PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9) → U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, X84, X85, T71, X12, X13, T9)
PP_IN_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9) → U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → PQ_IN_GGAGAGA(T71, T93, X12, T94, X13, T73, T9)
PQ_IN_GGAGAGA(T71, T93, T156, T94, X13, T73, T9) → QSA_IN_GA(cons(T71, T93), T156)

The TRS R consists of the following rules:

qsA_in_ga([], []) → qsA_out_ga([], [])
qsA_in_ga(cons(T14, []), T9) → U1_ga(T14, T9, pB_in_aaga(X12, X13, T14, T9))
pB_in_aaga(T19, X13, T14, T9) → U11_aaga(T19, X13, T14, T9, qsE_in_a(T19))
qsE_in_a([]) → qsE_out_a([])
U11_aaga(T19, X13, T14, T9, qsE_out_a(T19)) → U12_aaga(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
pO_in_agga(T24, T19, T14, T9) → U13_agga(T24, T19, T14, T9, qsE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, cons(T37, T38)) → appendF_out_ggga([], T37, T38, cons(T37, T38))
appendF_in_ggga(cons(T49, T50), T51, T52, cons(T49, T54)) → U4_ggga(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
U4_ggga(T49, T50, T51, T52, T54, appendF_out_ggga(T50, T51, T52, T54)) → appendF_out_ggga(cons(T49, T50), T51, T52, cons(T49, T54))
U14_agga(T24, T19, T14, T9, appendF_out_ggga(T19, T14, T24, T9)) → pO_out_agga(T24, T19, T14, T9)
U12_aaga(T19, X13, T14, T9, pO_out_agga(X13, T19, T14, T9)) → pB_out_aaga(T19, X13, T14, T9)
U1_ga(T14, T9, pB_out_aaga(X12, X13, T14, T9)) → qsA_out_ga(cons(T14, []), T9)
qsA_in_ga(cons(T73, cons(T71, T72)), T9) → U2_ga(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9) → U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_in_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9))
pP_in_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U17_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_in_ggagaga(T71, T93, X12, T94, X13, T73, T9))
pQ_in_ggagaga(T71, T93, T156, T94, X13, T73, T9) → U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_in_ga(cons(T71, T93), T156))
qsA_in_ga(cons(T214, cons(T212, T213)), T9) → U3_ga(T214, T212, T213, T9, pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9))
pD_in_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9) → U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_in_gg(T212, T214))
U33_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, geqK_out_gg(T212, T214)) → U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_in_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9))
pU_in_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9) → U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_in_ggaa(T213, T214, T221, T222))
U35_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, splitH_out_ggaa(T213, T214, T221, T222)) → U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_in_gaggaga(T221, X12, T212, T222, X13, T214, T9))
pV_in_gaggaga(T221, T225, T212, T222, X13, T214, T9) → U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_in_ga(T221, T225))
qsL_in_ga([], []) → qsL_out_ga([], [])
qsL_in_ga(cons(T162, T163), X217) → U9_ga(T162, T163, X217, pM_in_ggaaaaa(T163, T162, X213, X214, X215, X216, X217))
pM_in_ggaaaaa(T163, T162, T166, T167, X215, X216, X217) → U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_in_ggaa(T163, T162, T166, T167))
U27_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, splitH_out_ggaa(T163, T162, T166, T167)) → U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_in_gagaga(T166, X215, T167, X216, T162, X217))
pS_in_gagaga(T166, T170, T167, X216, T162, X217) → U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_in_ga(T166, T170))
U29_gagaga(T166, T170, T167, X216, T162, X217, qsL_out_ga(T166, T170)) → U30_gagaga(T166, T170, T167, X216, T162, X217, pT_in_gagga(T167, X216, T170, T162, X217))
pT_in_gagga(T167, T171, T170, T162, X217) → U31_gagga(T167, T171, T170, T162, X217, qsL_in_ga(T167, T171))
U31_gagga(T167, T171, T170, T162, X217, qsL_out_ga(T167, T171)) → U32_gagga(T167, T171, T170, T162, X217, appendN_in_ggga(T170, T162, T171, X217))
appendN_in_ggga([], T184, T185, cons(T184, T185)) → appendN_out_ggga([], T184, T185, cons(T184, T185))
appendN_in_ggga(cons(T194, T195), T196, T197, cons(T194, X243)) → U10_ggga(T194, T195, T196, T197, X243, appendN_in_ggga(T195, T196, T197, X243))
U10_ggga(T194, T195, T196, T197, X243, appendN_out_ggga(T195, T196, T197, X243)) → appendN_out_ggga(cons(T194, T195), T196, T197, cons(T194, X243))
U32_gagga(T167, T171, T170, T162, X217, appendN_out_ggga(T170, T162, T171, X217)) → pT_out_gagga(T167, T171, T170, T162, X217)
U30_gagaga(T166, T170, T167, X216, T162, X217, pT_out_gagga(T167, X216, T170, T162, X217)) → pS_out_gagaga(T166, T170, T167, X216, T162, X217)
U28_ggaaaaa(T163, T162, T166, T167, X215, X216, X217, pS_out_gagaga(T166, X215, T167, X216, T162, X217)) → pM_out_ggaaaaa(T163, T162, T166, T167, X215, X216, X217)
U9_ga(T162, T163, X217, pM_out_ggaaaaa(T163, T162, X213, X214, X215, X216, X217)) → qsL_out_ga(cons(T162, T163), X217)
U37_gaggaga(T221, T225, T212, T222, X13, T214, T9, qsL_out_ga(T221, T225)) → U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_in_gagga(cons(T212, T222), X13, T225, T214, T9))
pR_in_gagga(T94, T157, T156, T73, T9) → U25_gagga(T94, T157, T156, T73, T9, qsL_in_ga(T94, T157))
U25_gagga(T94, T157, T156, T73, T9, qsL_out_ga(T94, T157)) → U26_gagga(T94, T157, T156, T73, T9, appendF_in_ggga(T156, T73, T157, T9))
U26_gagga(T94, T157, T156, T73, T9, appendF_out_ggga(T156, T73, T157, T9)) → pR_out_gagga(T94, T157, T156, T73, T9)
U38_gaggaga(T221, T225, T212, T222, X13, T214, T9, pR_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → pV_out_gaggaga(T221, T225, T212, T222, X13, T214, T9)
U36_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9, pV_out_gaggaga(T221, X12, T212, T222, X13, T214, T9)) → pU_out_ggaaagaa(T213, T214, T221, T222, X12, T212, X13, T9)
U34_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9, pU_out_ggaaagaa(T213, T214, X267, X268, X12, T212, X13, T9)) → pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)
U3_ga(T214, T212, T213, T9, pD_out_gggaaaaa(T212, T214, T213, X267, X268, X12, X13, T9)) → qsA_out_ga(cons(T214, cons(T212, T213)), T9)
U19_ggagaga(T71, T93, T156, T94, X13, T73, T9, qsA_out_ga(cons(T71, T93), T156)) → U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_in_gagga(T94, X13, T156, T73, T9))
U20_ggagaga(T71, T93, T156, T94, X13, T73, T9, pR_out_gagga(T94, X13, T156, T73, T9)) → pQ_out_ggagaga(T71, T93, T156, T94, X13, T73, T9)
U18_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9, pQ_out_ggagaga(T71, T93, X12, T94, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T93, T94, T71, X12, X13, T9)
U16_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, pP_out_ggaagaaa(T72, T73, X84, X85, T71, X12, X13, T9)) → pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)
U2_ga(T73, T71, T72, T9, pC_out_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9)) → qsA_out_ga(cons(T73, cons(T71, T72)), T9)

(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(cons(T73, cons(T71, T72)), T9) → PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9)
PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9) → U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_in_gg(T71, T73))
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, lessG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, X84, X85, T71, X12, X13, T9)
PP_IN_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9) → U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T93, T94))
U17_GGAAGAAA(T72, T73, T93, T94, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T93, T94)) → PQ_IN_GGAGAGA(T71, T93, X12, T94, X13, T73, T9)
PQ_IN_GGAGAGA(T71, T93, T156, T94, X13, T73, T9) → QSA_IN_GA(cons(T71, T93), T156)

The TRS R consists of the following rules:

lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
splitH_in_ggaa([], T101, [], []) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116, cons(T114, X145), X146) → U6_ggaa(T114, T115, T116, X145, X146, pI_in_gggaa(T114, T116, T115, X145, X146))
splitH_in_ggaa(cons(T129, T130), T131, X172, cons(T129, X173)) → U7_ggaa(T129, T130, T131, X172, X173, pJ_in_gggaa(T129, T131, T130, X172, X173))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U6_ggaa(T114, T115, T116, X145, X146, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U7_ggaa(T129, T130, T131, X172, X173, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
pI_in_gggaa(T114, T116, T115, X145, X146) → U21_gggaa(T114, T116, T115, X145, X146, lessG_in_gg(T114, T116))
pJ_in_gggaa(T129, T131, T130, X172, X173) → U23_gggaa(T129, T131, T130, X172, X173, geqK_in_gg(T129, T131))
U21_gggaa(T114, T116, T115, X145, X146, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, X145, X146, splitH_in_ggaa(T115, T116, X145, X146))
U23_gggaa(T129, T131, T130, X172, X173, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, X172, X173, splitH_in_ggaa(T130, T131, X172, X173))
U22_gggaa(T114, T116, T115, X145, X146, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U24_gggaa(T129, T131, T130, X172, X173, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))

(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(cons(T73, cons(T71, T72))) → PC_IN_GGGAAAAA(T71, T73, T72)
PC_IN_GGGAAAAA(T71, T73, T72) → U15_GGGAAAAA(T71, T73, T72, lessG_in_gg(T71, T73))
U15_GGGAAAAA(T71, T73, T72, lessG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, T71)
PP_IN_GGAAGAAA(T72, T73, T71) → U17_GGAAGAAA(T72, T73, T71, splitH_in_ggaa(T72, T73))
U17_GGAAGAAA(T72, T73, T71, splitH_out_ggaa(T72, T73, T93, T94)) → PQ_IN_GGAGAGA(T71, T93, T94, T73)
PQ_IN_GGAGAGA(T71, T93, T94, T73) → QSA_IN_GA(cons(T71, T93))

The TRS R consists of the following rules:

lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
splitH_in_ggaa([], T101) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116) → U6_ggaa(T114, T115, T116, pI_in_gggaa(T114, T116, T115))
splitH_in_ggaa(cons(T129, T130), T131) → U7_ggaa(T129, T130, T131, pJ_in_gggaa(T129, T131, T130))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U6_ggaa(T114, T115, T116, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U7_ggaa(T129, T130, T131, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
pI_in_gggaa(T114, T116, T115) → U21_gggaa(T114, T116, T115, lessG_in_gg(T114, T116))
pJ_in_gggaa(T129, T131, T130) → U23_gggaa(T129, T131, T130, geqK_in_gg(T129, T131))
U21_gggaa(T114, T116, T115, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, splitH_in_ggaa(T115, T116))
U23_gggaa(T129, T131, T130, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, splitH_in_ggaa(T130, T131))
U22_gggaa(T114, T116, T115, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U24_gggaa(T129, T131, T130, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))

The set Q consists of the following terms:

lessG_in_gg(x0, x1)
splitH_in_ggaa(x0, x1)
U5_gg(x0, x1, x2)
U6_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
pI_in_gggaa(x0, x1, x2)
pJ_in_gggaa(x0, x1, x2)
U21_gggaa(x0, x1, x2, x3)
U23_gggaa(x0, x1, x2, x3)
U22_gggaa(x0, x1, x2, x3)
geqK_in_gg(x0, x1)
U24_gggaa(x0, x1, x2, x3)
U8_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.

QSA_IN_GA(cons(T73, cons(T71, T72))) → PC_IN_GGGAAAAA(T71, T73, T72)

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

POL( U15_GGGAAAAA(x₁, ..., x₄) ) = max{0, 2x₄ - 1}

POL( U17_GGAAGAAA(x₁, ..., x₄) ) = 2x₃ + 1

POL( U21_gggaa(x₁, ..., x₄) ) = max{0, -2}

POL( lessG_in_gg(x₁, x₂) ) = x₂

POL( 0 ) = 0

POL( s(x₁) ) = x₁ + 2

POL( lessG_out_gg(x₁, x₂) ) = x₁ + 1

POL( U5_gg(x₁, ..., x₃) ) = x₃ + 2

POL( splitH_in_ggaa(x₁, x₂) ) = 0

POL( [] ) = 1

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

POL( cons(x₁, x₂) ) = x₁

POL( U6_ggaa(x₁, ..., x₄) ) = 2x₂ + 2x₄ + 2

POL( pI_in_gggaa(x₁, ..., x₃) ) = 2

POL( U7_ggaa(x₁, ..., x₄) ) = 2x₃ + 2x₄ + 2

POL( pJ_in_gggaa(x₁, ..., x₃) ) = 0

POL( U22_gggaa(x₁, ..., x₄) ) = max{0, 2x₃ - 2}

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

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

POL( U23_gggaa(x₁, ..., x₄) ) = max{0, x₁ + x₃ - 2}

POL( geqK_in_gg(x₁, x₂) ) = x₂

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

POL( U8_gg(x₁, ..., x₃) ) = x₁ + 2x₂ + 2

POL( U24_gggaa(x₁, ..., x₄) ) = x₂ + x₃ + 2

POL( QSA_IN_GA(x₁) ) = 2x₁ + 1

POL( PC_IN_GGGAAAAA(x₁, ..., x₃) ) = 2x₂

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

POL( PQ_IN_GGAGAGA(x₁, ..., x₄) ) = 2x₁ + 1

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

lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))

(57) Obligation:

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

PC_IN_GGGAAAAA(T71, T73, T72) → U15_GGGAAAAA(T71, T73, T72, lessG_in_gg(T71, T73))
U15_GGGAAAAA(T71, T73, T72, lessG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, T71)
PP_IN_GGAAGAAA(T72, T73, T71) → U17_GGAAGAAA(T72, T73, T71, splitH_in_ggaa(T72, T73))
U17_GGAAGAAA(T72, T73, T71, splitH_out_ggaa(T72, T73, T93, T94)) → PQ_IN_GGAGAGA(T71, T93, T94, T73)
PQ_IN_GGAGAGA(T71, T93, T94, T73) → QSA_IN_GA(cons(T71, T93))

The TRS R consists of the following rules:

lessG_in_gg(0, s(T82)) → lessG_out_gg(0, s(T82))
lessG_in_gg(s(T87), s(T88)) → U5_gg(T87, T88, lessG_in_gg(T87, T88))
splitH_in_ggaa([], T101) → splitH_out_ggaa([], T101, [], [])
splitH_in_ggaa(cons(T114, T115), T116) → U6_ggaa(T114, T115, T116, pI_in_gggaa(T114, T116, T115))
splitH_in_ggaa(cons(T129, T130), T131) → U7_ggaa(T129, T130, T131, pJ_in_gggaa(T129, T131, T130))
U5_gg(T87, T88, lessG_out_gg(T87, T88)) → lessG_out_gg(s(T87), s(T88))
U6_ggaa(T114, T115, T116, pI_out_gggaa(T114, T116, T115, X145, X146)) → splitH_out_ggaa(cons(T114, T115), T116, cons(T114, X145), X146)
U7_ggaa(T129, T130, T131, pJ_out_gggaa(T129, T131, T130, X172, X173)) → splitH_out_ggaa(cons(T129, T130), T131, X172, cons(T129, X173))
pI_in_gggaa(T114, T116, T115) → U21_gggaa(T114, T116, T115, lessG_in_gg(T114, T116))
pJ_in_gggaa(T129, T131, T130) → U23_gggaa(T129, T131, T130, geqK_in_gg(T129, T131))
U21_gggaa(T114, T116, T115, lessG_out_gg(T114, T116)) → U22_gggaa(T114, T116, T115, splitH_in_ggaa(T115, T116))
U23_gggaa(T129, T131, T130, geqK_out_gg(T129, T131)) → U24_gggaa(T129, T131, T130, splitH_in_ggaa(T130, T131))
U22_gggaa(T114, T116, T115, splitH_out_ggaa(T115, T116, X145, X146)) → pI_out_gggaa(T114, T116, T115, X145, X146)
geqK_in_gg(T140, T140) → geqK_out_gg(T140, T140)
geqK_in_gg(s(T145), 0) → geqK_out_gg(s(T145), 0)
geqK_in_gg(s(T150), s(T151)) → U8_gg(T150, T151, geqK_in_gg(T150, T151))
U24_gggaa(T129, T131, T130, splitH_out_ggaa(T130, T131, X172, X173)) → pJ_out_gggaa(T129, T131, T130, X172, X173)
U8_gg(T150, T151, geqK_out_gg(T150, T151)) → geqK_out_gg(s(T150), s(T151))

The set Q consists of the following terms:

lessG_in_gg(x0, x1)
splitH_in_ggaa(x0, x1)
U5_gg(x0, x1, x2)
U6_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
pI_in_gggaa(x0, x1, x2)
pJ_in_gggaa(x0, x1, x2)
U21_gggaa(x0, x1, x2, x3)
U23_gggaa(x0, x1, x2, x3)
U22_gggaa(x0, x1, x2, x3)
geqK_in_gg(x0, x1)
U24_gggaa(x0, x1, x2, x3)
U8_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.

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (EQUIVALENT transformation)

(25) Obligation:

(26) QDPSizeChangeProof (EQUIVALENT transformation)

(27) YES

(28) Obligation:

(29) UsableRulesProof (EQUIVALENT transformation)

(30) Obligation:

(31) PiDPToQDPProof (SOUND transformation)

(32) Obligation:

(33) QDPSizeChangeProof (EQUIVALENT transformation)

(34) YES

(35) Obligation:

(36) UsableRulesProof (EQUIVALENT transformation)

(37) Obligation:

(38) PiDPToQDPProof (SOUND transformation)

(39) Obligation:

(40) QDPOrderProof (EQUIVALENT transformation)

(41) Obligation:

(42) DependencyGraphProof (EQUIVALENT transformation)

(43) TRUE

(44) Obligation:

(45) UsableRulesProof (EQUIVALENT transformation)

(46) Obligation:

(47) PiDPToQDPProof (SOUND transformation)

(48) Obligation:

(49) QDPSizeChangeProof (EQUIVALENT transformation)

(50) YES

(51) Obligation:

(52) UsableRulesProof (EQUIVALENT transformation)

(53) Obligation:

(54) PiDPToQDPProof (SOUND transformation)

(55) Obligation:

(56) QDPOrderProof (EQUIVALENT transformation)

(57) Obligation:

(58) DependencyGraphProof (EQUIVALENT transformation)

(59) TRUE