Left Termination proof of /tmp/tmpJONNbG/pl6.1.1.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 362 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 1646 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 106 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇒, 24 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 (⇒, 0 ms)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔, 0 ms)

        ↳34 YES

    ↳35 PiDP

        ↳36 UsableRulesProof (⇔, 0 ms)

        ↳37 PiDP

        ↳38 PiDPToQDPProof (⇒, 20 ms)

        ↳39 QDP

        ↳40 QDPOrderProof (⇔, 301 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 (⇔, 147 ms)

        ↳57 QDP

        ↳58 DependencyGraphProof (⇔, 0 ms)

        ↳59 TRUE

(0) Obligation:

Clauses:

qsort([], []).
qsort(.(H, L), S) :- ','(split(L, H, A, B), ','(qsort(A, A1), ','(qsort(B, B1), append(A1, .(H, B1), S)))).
split([], Y, [], []).
split(.(X, Xs), Y, .(X, Ls), Bs) :- ','(le(X, Y), split(Xs, Y, Ls, Bs)).
split(.(X, Xs), Y, Ls, .(X, Bs)) :- ','(gt(X, Y), split(Xs, Y, Ls, Bs)).
append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Query: qsort(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: qsort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: qsort(T1, T2), SCOPE: 1, CLAUSE: 0 | qsort(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true | qsort([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: qsort(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nqsort(T1, T2) !=? qsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: qsort([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(split(T7, T6, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(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];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(split(T7, T6, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(T6, X13), T9)))), SCOPE: 2, CLAUSE: 2 | ','(split(T7, T6, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(T6, X13), T9)))), SCOPE: 2, CLAUSE: 3 | ','(split(T7, T6, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(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];
29 [label="29: ','(split(T7, T6, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(T6, X13), T9)))), SCOPE: 2, CLAUSE: 2\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];
31 [label="31: ','(split(T7, T6, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(T6, X13), T9)))), SCOPE: 2, CLAUSE: 3 | ','(split(T7, T6, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(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];
34 [label="B: ','(qsort([], X12), ','(qsort([], X13), append(X12, .(T14, X13), T9)))\n\nT14 is ground\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="E: qsort([], X12)\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="O: ','(qsort([], X13), append(T19, .(T14, X13), T9))\n\nT14 is ground\nT19 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: qsort([], X12), SCOPE: 3, CLAUSE: 0 | qsort([], X12), SCOPE: 3, CLAUSE: 1\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: qsort([], X12), SCOPE: 3, CLAUSE: 0\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: qsort([], X12), SCOPE: 3, CLAUSE: 1\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: qsort([], X13)\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="F: append(T19, .(T14, T24), T9)\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: append(T19, .(T14, T24), T9), SCOPE: 4, CLAUSE: 5 | append(T19, .(T14, T24), T9), SCOPE: 4, CLAUSE: 6\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: append(T19, .(T14, T24), T9), SCOPE: 4, CLAUSE: 5\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: append(T19, .(T14, T24), T9), SCOPE: 4, CLAUSE: 6\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: append(T50, .(T51, T52), T54)\n\nT50 is ground\nT51 is ground\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(split(T7, T6, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(T6, X13), T9)))), SCOPE: 2, CLAUSE: 3\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];
96 [label="96: ','(split(T7, T6, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(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];
104 [label="C: ','(','(le(T71, T73), split(T72, T73, X84, X85)), ','(qsort(.(T71, X84), X12), ','(qsort(X85, X13), append(X12, .(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];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="G: le(T71, T73)\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="P: ','(split(T72, T73, X84, X85), ','(qsort(.(T71, X84), X12), ','(qsort(X85, X13), append(X12, .(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];
110 [label="110: le(T71, T73), SCOPE: 5, CLAUSE: 9 | le(T71, T73), SCOPE: 5, CLAUSE: 10 | le(T71, T73), SCOPE: 5, CLAUSE: 11\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: le(T71, T73), SCOPE: 5, CLAUSE: 9\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: le(T71, T73), SCOPE: 5, CLAUSE: 10 | le(T71, T73), SCOPE: 5, CLAUSE: 11\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: le(T86, T87)\n\nT86 is ground\nT87 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: le(T71, T73), SCOPE: 5, CLAUSE: 10\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: le(T71, T73), SCOPE: 5, CLAUSE: 11\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
137 [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];
138 [label="Q: ','(qsort(.(T71, T97), X12), ','(qsort(T98, X13), append(X12, .(T73, X13), T9)))\n\nT71 is ground\nT73 is ground\nT97 is ground\nT98 is ground\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 2 | split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 3 | 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];
145 [label="145: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 2\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 3 | 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];
152 [label="152: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 3\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: 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];
171 [label="I: ','(le(T118, T120), split(T119, T120, X149, X150))\n\nT118 is ground\nT119 is ground\nT120 is ground\nX149 is free\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
172 [label="172: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: le(T118, T120)\n\nT118 is ground\nT120 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: split(T119, T120, X149, X150)\n\nT119 is ground\nT120 is ground\nX149 is free\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
182 [label="J: ','(gt(T133, T135), split(T134, T135, X176, X177))\n\nT133 is ground\nT134 is ground\nT135 is ground\nX176 is free\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
185 [label="185: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="K: gt(T133, T135)\n\nT133 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
193 [label="193: split(T134, T135, X176, X177)\n\nT134 is ground\nT135 is ground\nX176 is free\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
197 [label="197: gt(T133, T135), SCOPE: 7, CLAUSE: 7 | gt(T133, T135), SCOPE: 7, CLAUSE: 8\n\nT133 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
198 [label="198: gt(T133, T135), SCOPE: 7, CLAUSE: 7\n\nT133 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: gt(T133, T135), SCOPE: 7, CLAUSE: 8\n\nT133 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: gt(T148, T149)\n\nT148 is ground\nT149 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="203: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
205 [label="205: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
206 [label="206: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
211 [label="211: qsort(.(T71, T97), X12)\n\nT71 is ground\nT97 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="R: ','(qsort(T98, X13), append(T157, .(T73, X13), T9))\n\nT73 is ground\nT98 is ground\nT157 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="L: qsort(T98, X13)\n\nT98 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
217 [label="217: append(T157, .(T73, T158), T9)\n\nT73 is ground\nT157 is ground\nT158 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
220 [label="220: qsort(T98, X13), SCOPE: 8, CLAUSE: 0 | qsort(T98, X13), SCOPE: 8, CLAUSE: 1\n\nT98 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: qsort(T98, X13), SCOPE: 8, CLAUSE: 0\n\nT98 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: qsort(T98, X13), SCOPE: 8, CLAUSE: 1\n\nT98 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
227 [label="227: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="228: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
233 [label="M: ','(split(T164, T163, X214, X215), ','(qsort(X214, X216), ','(qsort(X215, X217), append(X216, .(T163, X217), X218))))\n\nT163 is ground\nT164 is ground\nX218 is free\nX214 is free\nX215 is free\nX216 is free\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
234 [label="234: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
235 [label="235: split(T164, T163, X214, X215)\n\nT163 is ground\nT164 is ground\nX214 is free\nX215 is free", fontsize=16, style=filled, fillcolor=lightyellow];
236 [label="S: ','(qsort(T167, X216), ','(qsort(T168, X217), append(X216, .(T163, X217), X218)))\n\nT163 is ground\nT167 is ground\nT168 is ground\nX218 is free\nX216 is free\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
242 [label="242: qsort(T167, X216)\n\nT167 is ground\nX216 is free", fontsize=16, style=filled, fillcolor=lightyellow];
243 [label="T: ','(qsort(T168, X217), append(T171, .(T163, X217), X218))\n\nT163 is ground\nT168 is ground\nT171 is ground\nX218 is free\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
248 [label="248: qsort(T168, X217)\n\nT168 is ground\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
249 [label="N: append(T171, .(T163, T172), X218)\n\nT163 is ground\nT171 is ground\nT172 is ground\nX218 is free", fontsize=16, style=filled, fillcolor=lightyellow];
253 [label="253: append(T171, .(T163, T172), X218), SCOPE: 9, CLAUSE: 5 | append(T171, .(T163, T172), X218), SCOPE: 9, CLAUSE: 6\n\nT163 is ground\nT171 is ground\nT172 is ground\nX218 is free", fontsize=16, style=filled, fillcolor=lightyellow];
254 [label="254: append(T171, .(T163, T172), X218), SCOPE: 9, CLAUSE: 5\n\nT163 is ground\nT171 is ground\nT172 is ground\nX218 is free", fontsize=16, style=filled, fillcolor=lightyellow];
255 [label="255: append(T171, .(T163, T172), X218), SCOPE: 9, CLAUSE: 6\n\nT163 is ground\nT171 is ground\nT172 is ground\nX218 is free", fontsize=16, style=filled, fillcolor=lightyellow];
258 [label="258: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
259 [label="259: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
260 [label="260: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
273 [label="273: append(T196, .(T197, T198), X244)\n\nT196 is ground\nT197 is ground\nT198 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
274 [label="274: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
280 [label="D: ','(','(gt(T213, T215), split(T214, T215, X268, X269)), ','(qsort(X268, X12), ','(qsort(.(T213, X269), X13), append(X12, .(T215, X13), T9))))\n\nT213 is ground\nT214 is ground\nT215 is ground\nX12 is free\nX13 is free\nX268 is free\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
281 [label="281: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
289 [label="289: gt(T213, T215)\n\nT213 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
290 [label="U: ','(split(T214, T215, X268, X269), ','(qsort(X268, X12), ','(qsort(.(T213, X269), X13), append(X12, .(T215, X13), T9))))\n\nT213 is ground\nT214 is ground\nT215 is ground\nX12 is free\nX13 is free\nX268 is free\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
291 [label="291: split(T214, T215, X268, X269)\n\nT214 is ground\nT215 is ground\nX268 is free\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
292 [label="V: ','(qsort(T222, X12), ','(qsort(.(T213, T223), X13), append(X12, .(T215, X13), T9)))\n\nT213 is ground\nT215 is ground\nT222 is ground\nT223 is ground\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
297 [label="297: qsort(T222, X12)\n\nT222 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
298 [label="298: ','(qsort(.(T213, T223), X13), append(T226, .(T215, X13), T9))\n\nT213 is ground\nT215 is ground\nT223 is ground\nT226 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
299 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 299 [arrowhead = none ];
299 -> 6;

300 [label="EVAL with clause\nqsort([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 300 [arrowhead = none ];
300 -> 11;

301 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 301 [arrowhead = none ];
301 -> 14;

302 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 302 [arrowhead = none ];
302 -> 17;

303 [label="EVAL with clause\nqsort(.(X7, X8), X9) :- ','(split(X8, X7, X10, X11), ','(qsort(X10, X12), ','(qsort(X11, X13), append(X12, .(X7, X13), X9)))).\nand substitutionX7 -> T6,\nX8 -> T7,\nT1 -> .(T6, T7),\nT2 -> T9,\nX9 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 303 [arrowhead = none ];
303 -> 23;

304 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 304 [arrowhead = none ];
304 -> 25;

305 [label="BACKTRACK\nfor clause: qsort(.(H, L), S) :- ','(split(L, H, A, B), ','(qsort(A, A1), ','(qsort(B, B1), append(A1, .(H, B1), S))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 305 [arrowhead = none ];
305 -> 19;

306 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 306 [arrowhead = none ];
306 -> 27;

307 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 307 [arrowhead = none ];
307 -> 29;

308 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 308 [arrowhead = none ];
308 -> 31;

309 [label="EVAL with clause\nsplit([], X18, [], []).\nand substitutionT7 -> [],\nT6 -> T14,\nX18 -> T14,\nX10 -> [],\nX11 -> []", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 309 [arrowhead = none ];
309 -> 34;

310 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 310 [arrowhead = none ];
310 -> 35;

311 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 311 [arrowhead = none ];
311 -> 95;

312 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 312 [arrowhead = none ];
312 -> 96;

313 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
34 -> 313 [arrowhead = none ];
313 -> 38;

314 [label="SPLIT 2\nnew knowledge:\nT19 is ground\nreplacements:X12 -> T19", fontsize=14, style = filled, fillcolor = violet];
34 -> 314 [arrowhead = none ];
314 -> 39;

315 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 315 [arrowhead = none ];
315 -> 43;

316 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
39 -> 316 [arrowhead = none ];
316 -> 67;

317 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nreplacements:X13 -> T24", fontsize=14, style = filled, fillcolor = violet];
39 -> 317 [arrowhead = none ];
317 -> 68;

318 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 318 [arrowhead = none ];
318 -> 46;

319 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 319 [arrowhead = none ];
319 -> 47;

320 [label="ONLY EVAL with clause\nqsort([], []).\nand substitutionX12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 320 [arrowhead = none ];
320 -> 49;

321 [label="BACKTRACK\nfor clause: qsort(.(H, L), S) :- ','(split(L, H, A, B), ','(qsort(A, A1), ','(qsort(B, B1), append(A1, .(H, B1), S))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 321 [arrowhead = none ];
321 -> 59;

322 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 322 [arrowhead = none ];
322 -> 58;

323 [label="INSTANCE with matching:\nX12 -> X13", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 323 [arrowhead = none , style=dashed];
323 -> 38[style=dashed];

324 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 324 [arrowhead = none ];
324 -> 72;

325 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 325 [arrowhead = none ];
325 -> 73;

326 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 326 [arrowhead = none ];
326 -> 74;

327 [label="EVAL with clause\nappend([], X36, X36).\nand substitutionT19 -> [],\nT14 -> T37,\nT24 -> T38,\nX36 -> .(T37, T38),\nT9 -> .(T37, T38)", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 327 [arrowhead = none ];
327 -> 75;

328 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 328 [arrowhead = none ];
328 -> 77;

329 [label="EVAL with clause\nappend(.(X45, X46), X47, .(X45, X48)) :- append(X46, X47, X48).\nand substitutionX45 -> T49,\nX46 -> T50,\nT19 -> .(T49, T50),\nT14 -> T51,\nT24 -> T52,\nX47 -> .(T51, T52),\nX48 -> T54,\nT9 -> .(T49, T54),\nT53 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 329 [arrowhead = none ];
329 -> 87;

330 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 330 [arrowhead = none ];
330 -> 88;

331 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 331 [arrowhead = none ];
331 -> 78;

332 [label="INSTANCE with matching:\nT19 -> T50\nT14 -> T51\nT24 -> T52\nT9 -> T54", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 332 [arrowhead = none , style=dashed];
332 -> 68[style=dashed];

333 [label="EVAL with clause\nsplit(.(X79, X80), X81, .(X79, X82), X83) :- ','(le(X79, X81), split(X80, X81, X82, X83)).\nand substitutionX79 -> T71,\nX80 -> T72,\nT7 -> .(T71, T72),\nT6 -> T73,\nX81 -> T73,\nX82 -> X84,\nX10 -> .(T71, X84),\nX11 -> X85,\nX83 -> X85", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 333 [arrowhead = none ];
333 -> 104;

334 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 334 [arrowhead = none ];
334 -> 105;

335 [label="EVAL with clause\nsplit(.(X263, X264), X265, X266, .(X263, X267)) :- ','(gt(X263, X265), split(X264, X265, X266, X267)).\nand substitutionX263 -> T213,\nX264 -> T214,\nT7 -> .(T213, T214),\nT6 -> T215,\nX265 -> T215,\nX10 -> X268,\nX266 -> X268,\nX267 -> X269,\nX11 -> .(T213, X269)", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 335 [arrowhead = none ];
335 -> 280;

336 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 336 [arrowhead = none ];
336 -> 281;

337 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
104 -> 337 [arrowhead = none ];
337 -> 108;

338 [label="SPLIT 2\nnew knowledge:\nT71 is ground\nT73 is ground", fontsize=14, style = filled, fillcolor = violet];
104 -> 338 [arrowhead = none ];
338 -> 109;

339 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
108 -> 339 [arrowhead = none ];
339 -> 110;

340 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
109 -> 340 [arrowhead = none ];
340 -> 137;

341 [label="SPLIT 2\nnew knowledge:\nT72 is ground\nT73 is ground\nT97 is ground\nT98 is ground\nreplacements:X84 -> T97,\nX85 -> T98", fontsize=14, style = filled, fillcolor = violet];
109 -> 341 [arrowhead = none ];
341 -> 138;

342 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
110 -> 342 [arrowhead = none ];
342 -> 113;

343 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
110 -> 343 [arrowhead = none ];
343 -> 114;

344 [label="EVAL with clause\nle(s(X98), s(X99)) :- le(X98, X99).\nand substitutionX98 -> T86,\nT71 -> s(T86),\nX99 -> T87,\nT73 -> s(T87)", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 344 [arrowhead = none ];
344 -> 116;

345 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 345 [arrowhead = none ];
345 -> 117;

346 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
114 -> 346 [arrowhead = none ];
346 -> 122;

347 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
114 -> 347 [arrowhead = none ];
347 -> 123;

348 [label="INSTANCE with matching:\nT71 -> T86\nT73 -> T87", fontsize=14, style = filled, fillcolor = lightgrey];
116 -> 348 [arrowhead = none , style=dashed];
348 -> 108[style=dashed];

349 [label="EVAL with clause\nle(0, s(X106)).\nand substitutionT71 -> 0,\nX106 -> T94,\nT73 -> s(T94)", fontsize=14, style = filled, fillcolor = lightblue];
122 -> 349 [arrowhead = none ];
349 -> 125;

350 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
122 -> 350 [arrowhead = none ];
350 -> 126;

351 [label="EVAL with clause\nle(0, 0).\nand substitutionT71 -> 0,\nT73 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 351 [arrowhead = none ];
351 -> 130;

352 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
123 -> 352 [arrowhead = none ];
352 -> 131;

353 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
125 -> 353 [arrowhead = none ];
353 -> 128;

354 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
130 -> 354 [arrowhead = none ];
354 -> 132;

355 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
137 -> 355 [arrowhead = none ];
355 -> 141;

356 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
138 -> 356 [arrowhead = none ];
356 -> 211;

357 [label="SPLIT 2\nnew knowledge:\nT71 is ground\nT97 is ground\nT157 is ground\nreplacements:X12 -> T157", fontsize=14, style = filled, fillcolor = violet];
138 -> 357 [arrowhead = none ];
357 -> 212;

358 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
141 -> 358 [arrowhead = none ];
358 -> 145;

359 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
141 -> 359 [arrowhead = none ];
359 -> 146;

360 [label="EVAL with clause\nsplit([], X115, [], []).\nand substitutionT72 -> [],\nT73 -> T105,\nX115 -> T105,\nX84 -> [],\nX85 -> []", fontsize=14, style = filled, fillcolor = lightblue];
145 -> 360 [arrowhead = none ];
360 -> 152;

361 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
145 -> 361 [arrowhead = none ];
361 -> 153;

362 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
146 -> 362 [arrowhead = none ];
362 -> 159;

363 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
146 -> 363 [arrowhead = none ];
363 -> 160;

364 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
152 -> 364 [arrowhead = none ];
364 -> 154;

365 [label="EVAL with clause\nsplit(.(X144, X145), X146, .(X144, X147), X148) :- ','(le(X144, X146), split(X145, X146, X147, X148)).\nand substitutionX144 -> T118,\nX145 -> T119,\nT72 -> .(T118, T119),\nT73 -> T120,\nX146 -> T120,\nX147 -> X149,\nX84 -> .(T118, X149),\nX85 -> X150,\nX148 -> X150", fontsize=14, style = filled, fillcolor = lightblue];
159 -> 365 [arrowhead = none ];
365 -> 171;

366 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
159 -> 366 [arrowhead = none ];
366 -> 172;

367 [label="EVAL with clause\nsplit(.(X171, X172), X173, X174, .(X171, X175)) :- ','(gt(X171, X173), split(X172, X173, X174, X175)).\nand substitutionX171 -> T133,\nX172 -> T134,\nT72 -> .(T133, T134),\nT73 -> T135,\nX173 -> T135,\nX84 -> X176,\nX174 -> X176,\nX175 -> X177,\nX85 -> .(T133, X177)", fontsize=14, style = filled, fillcolor = lightblue];
160 -> 367 [arrowhead = none ];
367 -> 182;

368 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
160 -> 368 [arrowhead = none ];
368 -> 185;

369 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
171 -> 369 [arrowhead = none ];
369 -> 173;

370 [label="SPLIT 2\nnew knowledge:\nT118 is ground\nT120 is ground", fontsize=14, style = filled, fillcolor = violet];
171 -> 370 [arrowhead = none ];
370 -> 174;

371 [label="INSTANCE with matching:\nT71 -> T118\nT73 -> T120", fontsize=14, style = filled, fillcolor = lightgrey];
173 -> 371 [arrowhead = none , style=dashed];
371 -> 108[style=dashed];

372 [label="INSTANCE with matching:\nT72 -> T119\nT73 -> T120\nX84 -> X149\nX85 -> X150", fontsize=14, style = filled, fillcolor = lightgrey];
174 -> 372 [arrowhead = none , style=dashed];
372 -> 137[style=dashed];

373 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
182 -> 373 [arrowhead = none ];
373 -> 192;

374 [label="SPLIT 2\nnew knowledge:\nT133 is ground\nT135 is ground", fontsize=14, style = filled, fillcolor = violet];
182 -> 374 [arrowhead = none ];
374 -> 193;

375 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
192 -> 375 [arrowhead = none ];
375 -> 197;

376 [label="INSTANCE with matching:\nT72 -> T134\nT73 -> T135\nX84 -> X176\nX85 -> X177", fontsize=14, style = filled, fillcolor = lightgrey];
193 -> 376 [arrowhead = none , style=dashed];
376 -> 137[style=dashed];

377 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
197 -> 377 [arrowhead = none ];
377 -> 198;

378 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
197 -> 378 [arrowhead = none ];
378 -> 199;

379 [label="EVAL with clause\ngt(s(X190), s(X191)) :- gt(X190, X191).\nand substitutionX190 -> T148,\nT133 -> s(T148),\nX191 -> T149,\nT135 -> s(T149)", fontsize=14, style = filled, fillcolor = lightblue];
198 -> 379 [arrowhead = none ];
379 -> 202;

380 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
198 -> 380 [arrowhead = none ];
380 -> 203;

381 [label="EVAL with clause\ngt(s(X196), 0).\nand substitutionX196 -> T154,\nT133 -> s(T154),\nT135 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
199 -> 381 [arrowhead = none ];
381 -> 204;

382 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
199 -> 382 [arrowhead = none ];
382 -> 205;

383 [label="INSTANCE with matching:\nT133 -> T148\nT135 -> T149", fontsize=14, style = filled, fillcolor = lightgrey];
202 -> 383 [arrowhead = none , style=dashed];
383 -> 192[style=dashed];

384 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
204 -> 384 [arrowhead = none ];
384 -> 206;

385 [label="INSTANCE with matching:\nT1 -> .(T71, T97)\nT2 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
211 -> 385 [arrowhead = none , style=dashed];
385 -> 2[style=dashed];

386 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
212 -> 386 [arrowhead = none ];
386 -> 216;

387 [label="SPLIT 2\nnew knowledge:\nT98 is ground\nT158 is ground\nreplacements:X13 -> T158", fontsize=14, style = filled, fillcolor = violet];
212 -> 387 [arrowhead = none ];
387 -> 217;

388 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
216 -> 388 [arrowhead = none ];
388 -> 220;

389 [label="INSTANCE with matching:\nT19 -> T157\nT14 -> T73\nT24 -> T158", fontsize=14, style = filled, fillcolor = lightgrey];
217 -> 389 [arrowhead = none , style=dashed];
389 -> 68[style=dashed];

390 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
220 -> 390 [arrowhead = none ];
390 -> 222;

391 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
220 -> 391 [arrowhead = none ];
391 -> 223;

392 [label="EVAL with clause\nqsort([], []).\nand substitutionT98 -> [],\nX13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
222 -> 392 [arrowhead = none ];
392 -> 226;

393 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
222 -> 393 [arrowhead = none ];
393 -> 227;

394 [label="EVAL with clause\nqsort(.(X211, X212), X213) :- ','(split(X212, X211, X214, X215), ','(qsort(X214, X216), ','(qsort(X215, X217), append(X216, .(X211, X217), X213)))).\nand substitutionX211 -> T163,\nX212 -> T164,\nT98 -> .(T163, T164),\nX13 -> X218,\nX213 -> X218", fontsize=14, style = filled, fillcolor = lightblue];
223 -> 394 [arrowhead = none ];
394 -> 233;

395 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
223 -> 395 [arrowhead = none ];
395 -> 234;

396 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
226 -> 396 [arrowhead = none ];
396 -> 228;

397 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
233 -> 397 [arrowhead = none ];
397 -> 235;

398 [label="SPLIT 2\nnew knowledge:\nT164 is ground\nT163 is ground\nT167 is ground\nT168 is ground\nreplacements:X214 -> T167,\nX215 -> T168", fontsize=14, style = filled, fillcolor = violet];
233 -> 398 [arrowhead = none ];
398 -> 236;

399 [label="INSTANCE with matching:\nT72 -> T164\nT73 -> T163\nX84 -> X214\nX85 -> X215", fontsize=14, style = filled, fillcolor = lightgrey];
235 -> 399 [arrowhead = none , style=dashed];
399 -> 137[style=dashed];

400 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
236 -> 400 [arrowhead = none ];
400 -> 242;

401 [label="SPLIT 2\nnew knowledge:\nT167 is ground\nT171 is ground\nreplacements:X216 -> T171", fontsize=14, style = filled, fillcolor = violet];
236 -> 401 [arrowhead = none ];
401 -> 243;

402 [label="INSTANCE with matching:\nT98 -> T167\nX13 -> X216", fontsize=14, style = filled, fillcolor = lightgrey];
242 -> 402 [arrowhead = none , style=dashed];
402 -> 216[style=dashed];

403 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
243 -> 403 [arrowhead = none ];
403 -> 248;

404 [label="SPLIT 2\nnew knowledge:\nT168 is ground\nT172 is ground\nreplacements:X217 -> T172", fontsize=14, style = filled, fillcolor = violet];
243 -> 404 [arrowhead = none ];
404 -> 249;

405 [label="INSTANCE with matching:\nT98 -> T168\nX13 -> X217", fontsize=14, style = filled, fillcolor = lightgrey];
248 -> 405 [arrowhead = none , style=dashed];
405 -> 216[style=dashed];

406 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
249 -> 406 [arrowhead = none ];
406 -> 253;

407 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
253 -> 407 [arrowhead = none ];
407 -> 254;

408 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
253 -> 408 [arrowhead = none ];
408 -> 255;

409 [label="EVAL with clause\nappend([], X229, X229).\nand substitutionT171 -> [],\nT163 -> T185,\nT172 -> T186,\nX229 -> .(T185, T186),\nX218 -> .(T185, T186)", fontsize=14, style = filled, fillcolor = lightblue];
254 -> 409 [arrowhead = none ];
409 -> 258;

410 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
254 -> 410 [arrowhead = none ];
410 -> 259;

411 [label="EVAL with clause\nappend(.(X240, X241), X242, .(X240, X243)) :- append(X241, X242, X243).\nand substitutionX240 -> T195,\nX241 -> T196,\nT171 -> .(T195, T196),\nT163 -> T197,\nT172 -> T198,\nX242 -> .(T197, T198),\nX243 -> X244,\nX218 -> .(T195, X244)", fontsize=14, style = filled, fillcolor = lightblue];
255 -> 411 [arrowhead = none ];
411 -> 273;

412 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
255 -> 412 [arrowhead = none ];
412 -> 274;

413 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
258 -> 413 [arrowhead = none ];
413 -> 260;

414 [label="INSTANCE with matching:\nT171 -> T196\nT163 -> T197\nT172 -> T198\nX218 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
273 -> 414 [arrowhead = none , style=dashed];
414 -> 249[style=dashed];

415 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
280 -> 415 [arrowhead = none ];
415 -> 289;

416 [label="SPLIT 2\nnew knowledge:\nT213 is ground\nT215 is ground", fontsize=14, style = filled, fillcolor = violet];
280 -> 416 [arrowhead = none ];
416 -> 290;

417 [label="INSTANCE with matching:\nT133 -> T213\nT135 -> T215", fontsize=14, style = filled, fillcolor = lightgrey];
289 -> 417 [arrowhead = none , style=dashed];
417 -> 192[style=dashed];

418 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
290 -> 418 [arrowhead = none ];
418 -> 291;

419 [label="SPLIT 2\nnew knowledge:\nT214 is ground\nT215 is ground\nT222 is ground\nT223 is ground\nreplacements:X268 -> T222,\nX269 -> T223", fontsize=14, style = filled, fillcolor = violet];
290 -> 419 [arrowhead = none ];
419 -> 292;

420 [label="INSTANCE with matching:\nT72 -> T214\nT73 -> T215\nX84 -> X268\nX85 -> X269", fontsize=14, style = filled, fillcolor = lightgrey];
291 -> 420 [arrowhead = none , style=dashed];
420 -> 137[style=dashed];

421 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
292 -> 421 [arrowhead = none ];
421 -> 297;

422 [label="SPLIT 2\nnew knowledge:\nT222 is ground\nT226 is ground\nreplacements:X12 -> T226", fontsize=14, style = filled, fillcolor = violet];
292 -> 422 [arrowhead = none ];
422 -> 298;

423 [label="INSTANCE with matching:\nT98 -> T222\nX13 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
297 -> 423 [arrowhead = none , style=dashed];
423 -> 216[style=dashed];

424 [label="INSTANCE with matching:\nT98 -> .(T213, T223)\nT157 -> T226\nT73 -> T215", fontsize=14, style = filled, fillcolor = lightgrey];
298 -> 424 [arrowhead = none , style=dashed];
424 -> 212[style=dashed];

}

(2) Obligation:

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

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(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:

QSORTA_IN_GA(.(T14, []), T9) → U1_GA(T14, T9, pB_in_aaga(X12, X13, T14, T9))
QSORTA_IN_GA(.(T14, []), T9) → PB_IN_AAGA(X12, X13, T14, T9)
PB_IN_AAGA(T19, X13, T14, T9) → U11_AAGA(T19, X13, T14, T9, qsortE_in_a(T19))
PB_IN_AAGA(T19, X13, T14, T9) → QSORTE_IN_A(T19)
U11_AAGA(T19, X13, T14, T9, qsortE_out_a(T19)) → U12_AAGA(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
U11_AAGA(T19, X13, T14, T9, qsortE_out_a(T19)) → PO_IN_AGGA(X13, T19, T14, T9)
PO_IN_AGGA(T24, T19, T14, T9) → U13_AGGA(T24, T19, T14, T9, qsortE_in_a(T24))
PO_IN_AGGA(T24, T19, T14, T9) → QSORTE_IN_A(T24)
U13_AGGA(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_AGGA(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
U13_AGGA(T24, T19, T14, T9, qsortE_out_a(T24)) → APPENDF_IN_GGGA(T19, T14, T24, T9)
APPENDF_IN_GGGA(.(T49, T50), T51, T52, .(T49, T54)) → U4_GGGA(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
APPENDF_IN_GGGA(.(T49, T50), T51, T52, .(T49, T54)) → APPENDF_IN_GGGA(T50, T51, T52, T54)
QSORTA_IN_GA(.(T73, .(T71, T72)), T9) → U2_GA(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
QSORTA_IN_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, leG_in_gg(T71, T73))
PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9) → LEG_IN_GG(T71, T73)
LEG_IN_GG(s(T86), s(T87)) → U5_GG(T86, T87, leG_in_gg(T86, T87))
LEG_IN_GG(s(T86), s(T87)) → LEG_IN_GG(T86, T87)
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, leG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, X84, X85, T71, X12, X13, T9)
PP_IN_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9) → U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
PP_IN_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9) → SPLITH_IN_GGAA(T72, T73, T97, T98)
SPLITH_IN_GGAA(.(T118, T119), T120, .(T118, X149), X150) → U6_GGAA(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
SPLITH_IN_GGAA(.(T118, T119), T120, .(T118, X149), X150) → PI_IN_GGGAA(T118, T120, T119, X149, X150)
PI_IN_GGGAA(T118, T120, T119, X149, X150) → U21_GGGAA(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
PI_IN_GGGAA(T118, T120, T119, X149, X150) → LEG_IN_GG(T118, T120)
U21_GGGAA(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_GGGAA(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
U21_GGGAA(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → SPLITH_IN_GGAA(T119, T120, X149, X150)
SPLITH_IN_GGAA(.(T133, T134), T135, X176, .(T133, X177)) → U7_GGAA(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
SPLITH_IN_GGAA(.(T133, T134), T135, X176, .(T133, X177)) → PJ_IN_GGGAA(T133, T135, T134, X176, X177)
PJ_IN_GGGAA(T133, T135, T134, X176, X177) → U23_GGGAA(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
PJ_IN_GGGAA(T133, T135, T134, X176, X177) → GTK_IN_GG(T133, T135)
GTK_IN_GG(s(T148), s(T149)) → U8_GG(T148, T149, gtK_in_gg(T148, T149))
GTK_IN_GG(s(T148), s(T149)) → GTK_IN_GG(T148, T149)
U23_GGGAA(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_GGGAA(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U23_GGGAA(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → SPLITH_IN_GGAA(T134, T135, X176, X177)
U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → PQ_IN_GGAGAGA(T71, T97, X12, T98, X13, T73, T9)
PQ_IN_GGAGAGA(T71, T97, T157, T98, X13, T73, T9) → U19_GGAGAGA(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
PQ_IN_GGAGAGA(T71, T97, T157, T98, X13, T73, T9) → QSORTA_IN_GA(.(T71, T97), T157)
QSORTA_IN_GA(.(T215, .(T213, T214)), T9) → U3_GA(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
QSORTA_IN_GA(.(T215, .(T213, T214)), T9) → PD_IN_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9)
PD_IN_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9) → U33_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
PD_IN_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9) → GTK_IN_GG(T213, T215)
U33_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
U33_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → PU_IN_GGAAAGAA(T214, T215, X268, X269, X12, T213, X13, T9)
PU_IN_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9) → U35_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
PU_IN_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9) → SPLITH_IN_GGAA(T214, T215, T222, T223)
U35_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
U35_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → PV_IN_GAGGAGA(T222, X12, T213, T223, X13, T215, T9)
PV_IN_GAGGAGA(T222, T226, T213, T223, X13, T215, T9) → U37_GAGGAGA(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
PV_IN_GAGGAGA(T222, T226, T213, T223, X13, T215, T9) → QSORTL_IN_GA(T222, T226)
QSORTL_IN_GA(.(T163, T164), X218) → U9_GA(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
QSORTL_IN_GA(.(T163, T164), X218) → PM_IN_GGAAAAA(T164, T163, X214, X215, X216, X217, X218)
PM_IN_GGAAAAA(T164, T163, T167, T168, X216, X217, X218) → U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
PM_IN_GGAAAAA(T164, T163, T167, T168, X216, X217, X218) → SPLITH_IN_GGAA(T164, T163, T167, T168)
U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → PS_IN_GAGAGA(T167, X216, T168, X217, T163, X218)
PS_IN_GAGAGA(T167, T171, T168, X217, T163, X218) → U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
PS_IN_GAGAGA(T167, T171, T168, X217, T163, X218) → QSORTL_IN_GA(T167, T171)
U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_GAGAGA(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → PT_IN_GAGGA(T168, X217, T171, T163, X218)
PT_IN_GAGGA(T168, T172, T171, T163, X218) → U31_GAGGA(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
PT_IN_GAGGA(T168, T172, T171, T163, X218) → QSORTL_IN_GA(T168, T172)
U31_GAGGA(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_GAGGA(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
U31_GAGGA(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → APPENDN_IN_GGGA(T171, T163, T172, X218)
APPENDN_IN_GGGA(.(T195, T196), T197, T198, .(T195, X244)) → U10_GGGA(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
APPENDN_IN_GGGA(.(T195, T196), T197, T198, .(T195, X244)) → APPENDN_IN_GGGA(T196, T197, T198, X244)
U37_GAGGAGA(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_GAGGAGA(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
U37_GAGGAGA(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → PR_IN_GAGGA(.(T213, T223), X13, T226, T215, T9)
PR_IN_GAGGA(T98, T158, T157, T73, T9) → U25_GAGGA(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
PR_IN_GAGGA(T98, T158, T157, T73, T9) → QSORTL_IN_GA(T98, T158)
U25_GAGGA(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_GAGGA(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U25_GAGGA(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → APPENDF_IN_GGGA(T157, T73, T158, T9)
U19_GGAGAGA(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_GGAGAGA(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U19_GGAGAGA(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → PR_IN_GAGGA(T98, X13, T157, T73, T9)

The TRS R consists of the following rules:

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(T71, T72)), T9)

The argument filtering Pi contains the following mapping:
qsortA_in_ga(x1, x2) = qsortA_in_ga(x1)
[] = []
qsortA_out_ga(x1, x2) = qsortA_out_ga(x1, x2)
.(x1, x2) = .(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)
qsortE_in_a(x1) = qsortE_in_a
qsortE_out_a(x1) = qsortE_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)
leG_in_gg(x1, x2) = leG_in_gg(x1, x2)
s(x1) = s(x1)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
0 = 0
leG_out_gg(x1, x2) = leG_out_gg(x1, x2)
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)
gtK_in_gg(x1, x2) = gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2) = gtK_out_gg(x1, x2)
U24_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)
qsortL_in_ga(x1, x2) = qsortL_in_ga(x1)
qsortL_out_ga(x1, x2) = qsortL_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)
QSORTA_IN_GA(x1, x2) = QSORTA_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)
QSORTE_IN_A(x1) = QSORTE_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)
LEG_IN_GG(x1, x2) = LEG_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)
GTK_IN_GG(x1, x2) = GTK_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)
QSORTL_IN_GA(x1, x2) = QSORTL_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:

QSORTA_IN_GA(.(T14, []), T9) → U1_GA(T14, T9, pB_in_aaga(X12, X13, T14, T9))
QSORTA_IN_GA(.(T14, []), T9) → PB_IN_AAGA(X12, X13, T14, T9)
PB_IN_AAGA(T19, X13, T14, T9) → U11_AAGA(T19, X13, T14, T9, qsortE_in_a(T19))
PB_IN_AAGA(T19, X13, T14, T9) → QSORTE_IN_A(T19)
U11_AAGA(T19, X13, T14, T9, qsortE_out_a(T19)) → U12_AAGA(T19, X13, T14, T9, pO_in_agga(X13, T19, T14, T9))
U11_AAGA(T19, X13, T14, T9, qsortE_out_a(T19)) → PO_IN_AGGA(X13, T19, T14, T9)
PO_IN_AGGA(T24, T19, T14, T9) → U13_AGGA(T24, T19, T14, T9, qsortE_in_a(T24))
PO_IN_AGGA(T24, T19, T14, T9) → QSORTE_IN_A(T24)
U13_AGGA(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_AGGA(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
U13_AGGA(T24, T19, T14, T9, qsortE_out_a(T24)) → APPENDF_IN_GGGA(T19, T14, T24, T9)
APPENDF_IN_GGGA(.(T49, T50), T51, T52, .(T49, T54)) → U4_GGGA(T49, T50, T51, T52, T54, appendF_in_ggga(T50, T51, T52, T54))
APPENDF_IN_GGGA(.(T49, T50), T51, T52, .(T49, T54)) → APPENDF_IN_GGGA(T50, T51, T52, T54)
QSORTA_IN_GA(.(T73, .(T71, T72)), T9) → U2_GA(T73, T71, T72, T9, pC_in_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9))
QSORTA_IN_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, leG_in_gg(T71, T73))
PC_IN_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9) → LEG_IN_GG(T71, T73)
LEG_IN_GG(s(T86), s(T87)) → U5_GG(T86, T87, leG_in_gg(T86, T87))
LEG_IN_GG(s(T86), s(T87)) → LEG_IN_GG(T86, T87)
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, leG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, X84, X85, T71, X12, X13, T9)
PP_IN_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9) → U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
PP_IN_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9) → SPLITH_IN_GGAA(T72, T73, T97, T98)
SPLITH_IN_GGAA(.(T118, T119), T120, .(T118, X149), X150) → U6_GGAA(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
SPLITH_IN_GGAA(.(T118, T119), T120, .(T118, X149), X150) → PI_IN_GGGAA(T118, T120, T119, X149, X150)
PI_IN_GGGAA(T118, T120, T119, X149, X150) → U21_GGGAA(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
PI_IN_GGGAA(T118, T120, T119, X149, X150) → LEG_IN_GG(T118, T120)
U21_GGGAA(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_GGGAA(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
U21_GGGAA(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → SPLITH_IN_GGAA(T119, T120, X149, X150)
SPLITH_IN_GGAA(.(T133, T134), T135, X176, .(T133, X177)) → U7_GGAA(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
SPLITH_IN_GGAA(.(T133, T134), T135, X176, .(T133, X177)) → PJ_IN_GGGAA(T133, T135, T134, X176, X177)
PJ_IN_GGGAA(T133, T135, T134, X176, X177) → U23_GGGAA(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
PJ_IN_GGGAA(T133, T135, T134, X176, X177) → GTK_IN_GG(T133, T135)
GTK_IN_GG(s(T148), s(T149)) → U8_GG(T148, T149, gtK_in_gg(T148, T149))
GTK_IN_GG(s(T148), s(T149)) → GTK_IN_GG(T148, T149)
U23_GGGAA(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_GGGAA(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U23_GGGAA(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → SPLITH_IN_GGAA(T134, T135, X176, X177)
U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → PQ_IN_GGAGAGA(T71, T97, X12, T98, X13, T73, T9)
PQ_IN_GGAGAGA(T71, T97, T157, T98, X13, T73, T9) → U19_GGAGAGA(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
PQ_IN_GGAGAGA(T71, T97, T157, T98, X13, T73, T9) → QSORTA_IN_GA(.(T71, T97), T157)
QSORTA_IN_GA(.(T215, .(T213, T214)), T9) → U3_GA(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
QSORTA_IN_GA(.(T215, .(T213, T214)), T9) → PD_IN_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9)
PD_IN_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9) → U33_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
PD_IN_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9) → GTK_IN_GG(T213, T215)
U33_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
U33_GGGAAAAA(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → PU_IN_GGAAAGAA(T214, T215, X268, X269, X12, T213, X13, T9)
PU_IN_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9) → U35_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
PU_IN_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9) → SPLITH_IN_GGAA(T214, T215, T222, T223)
U35_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
U35_GGAAAGAA(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → PV_IN_GAGGAGA(T222, X12, T213, T223, X13, T215, T9)
PV_IN_GAGGAGA(T222, T226, T213, T223, X13, T215, T9) → U37_GAGGAGA(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
PV_IN_GAGGAGA(T222, T226, T213, T223, X13, T215, T9) → QSORTL_IN_GA(T222, T226)
QSORTL_IN_GA(.(T163, T164), X218) → U9_GA(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
QSORTL_IN_GA(.(T163, T164), X218) → PM_IN_GGAAAAA(T164, T163, X214, X215, X216, X217, X218)
PM_IN_GGAAAAA(T164, T163, T167, T168, X216, X217, X218) → U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
PM_IN_GGAAAAA(T164, T163, T167, T168, X216, X217, X218) → SPLITH_IN_GGAA(T164, T163, T167, T168)
U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → PS_IN_GAGAGA(T167, X216, T168, X217, T163, X218)
PS_IN_GAGAGA(T167, T171, T168, X217, T163, X218) → U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
PS_IN_GAGAGA(T167, T171, T168, X217, T163, X218) → QSORTL_IN_GA(T167, T171)
U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_GAGAGA(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → PT_IN_GAGGA(T168, X217, T171, T163, X218)
PT_IN_GAGGA(T168, T172, T171, T163, X218) → U31_GAGGA(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
PT_IN_GAGGA(T168, T172, T171, T163, X218) → QSORTL_IN_GA(T168, T172)
U31_GAGGA(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_GAGGA(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
U31_GAGGA(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → APPENDN_IN_GGGA(T171, T163, T172, X218)
APPENDN_IN_GGGA(.(T195, T196), T197, T198, .(T195, X244)) → U10_GGGA(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
APPENDN_IN_GGGA(.(T195, T196), T197, T198, .(T195, X244)) → APPENDN_IN_GGGA(T196, T197, T198, X244)
U37_GAGGAGA(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_GAGGAGA(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
U37_GAGGAGA(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → PR_IN_GAGGA(.(T213, T223), X13, T226, T215, T9)
PR_IN_GAGGA(T98, T158, T157, T73, T9) → U25_GAGGA(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
PR_IN_GAGGA(T98, T158, T157, T73, T9) → QSORTL_IN_GA(T98, T158)
U25_GAGGA(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_GAGGA(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U25_GAGGA(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → APPENDF_IN_GGGA(T157, T73, T158, T9)
U19_GGAGAGA(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_GGAGAGA(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U19_GGAGAGA(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → PR_IN_GAGGA(T98, X13, T157, T73, T9)

The TRS R consists of the following rules:

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(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(.(T195, T196), T197, T198, .(T195, X244)) → APPENDN_IN_GGGA(T196, T197, T198, X244)

The TRS R consists of the following rules:

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(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(.(T195, T196), T197, T198, .(T195, X244)) → APPENDN_IN_GGGA(T196, T197, T198, X244)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(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(.(T195, T196), T197, T198) → APPENDN_IN_GGGA(T196, T197, T198)

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(.(T195, T196), T197, T198) → APPENDN_IN_GGGA(T196, T197, T198)
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:

GTK_IN_GG(s(T148), s(T149)) → GTK_IN_GG(T148, T149)

The TRS R consists of the following rules:

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(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:

GTK_IN_GG(s(T148), s(T149)) → GTK_IN_GG(T148, T149)

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:

GTK_IN_GG(s(T148), s(T149)) → GTK_IN_GG(T148, T149)

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:

GTK_IN_GG(s(T148), s(T149)) → GTK_IN_GG(T148, T149)
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:

LEG_IN_GG(s(T86), s(T87)) → LEG_IN_GG(T86, T87)

The TRS R consists of the following rules:

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(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:

LEG_IN_GG(s(T86), s(T87)) → LEG_IN_GG(T86, T87)

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:

LEG_IN_GG(s(T86), s(T87)) → LEG_IN_GG(T86, T87)

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:

LEG_IN_GG(s(T86), s(T87)) → LEG_IN_GG(T86, T87)
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(T118, T120, T119, X149, X150) → U21_GGGAA(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_GGGAA(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → SPLITH_IN_GGAA(T119, T120, X149, X150)
SPLITH_IN_GGAA(.(T118, T119), T120, .(T118, X149), X150) → PI_IN_GGGAA(T118, T120, T119, X149, X150)
SPLITH_IN_GGAA(.(T133, T134), T135, X176, .(T133, X177)) → PJ_IN_GGGAA(T133, T135, T134, X176, X177)
PJ_IN_GGGAA(T133, T135, T134, X176, X177) → U23_GGGAA(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
U23_GGGAA(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → SPLITH_IN_GGAA(T134, T135, X176, X177)

The TRS R consists of the following rules:

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(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(T118, T120, T119, X149, X150) → U21_GGGAA(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_GGGAA(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → SPLITH_IN_GGAA(T119, T120, X149, X150)
SPLITH_IN_GGAA(.(T118, T119), T120, .(T118, X149), X150) → PI_IN_GGGAA(T118, T120, T119, X149, X150)
SPLITH_IN_GGAA(.(T133, T134), T135, X176, .(T133, X177)) → PJ_IN_GGGAA(T133, T135, T134, X176, X177)
PJ_IN_GGGAA(T133, T135, T134, X176, X177) → U23_GGGAA(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
U23_GGGAA(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → SPLITH_IN_GGAA(T134, T135, X176, X177)

The TRS R consists of the following rules:

leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
leG_in_gg(x1, x2) = leG_in_gg(x1, x2)
s(x1) = s(x1)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
0 = 0
leG_out_gg(x1, x2) = leG_out_gg(x1, x2)
gtK_in_gg(x1, x2) = gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2) = gtK_out_gg(x1, x2)
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(T118, T120, T119) → U21_GGGAA(T118, T120, T119, leG_in_gg(T118, T120))
U21_GGGAA(T118, T120, T119, leG_out_gg(T118, T120)) → SPLITH_IN_GGAA(T119, T120)
SPLITH_IN_GGAA(.(T118, T119), T120) → PI_IN_GGGAA(T118, T120, T119)
SPLITH_IN_GGAA(.(T133, T134), T135) → PJ_IN_GGGAA(T133, T135, T134)
PJ_IN_GGGAA(T133, T135, T134) → U23_GGGAA(T133, T135, T134, gtK_in_gg(T133, T135))
U23_GGGAA(T133, T135, T134, gtK_out_gg(T133, T135)) → SPLITH_IN_GGAA(T134, T135)

The TRS R consists of the following rules:

leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))

The set Q consists of the following terms:

leG_in_gg(x0, x1)
gtK_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(T118, T120, T119, leG_out_gg(T118, T120)) → SPLITH_IN_GGAA(T119, T120)
The graph contains the following edges 3 >= 1, 2 >= 2, 4 > 2
SPLITH_IN_GGAA(.(T118, T119), T120) → PI_IN_GGGAA(T118, T120, T119)
The graph contains the following edges 1 > 1, 2 >= 2, 1 > 3
SPLITH_IN_GGAA(.(T133, T134), T135) → PJ_IN_GGGAA(T133, T135, T134)
The graph contains the following edges 1 > 1, 2 >= 2, 1 > 3
PI_IN_GGGAA(T118, T120, T119) → U21_GGGAA(T118, T120, T119, leG_in_gg(T118, T120))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
U23_GGGAA(T133, T135, T134, gtK_out_gg(T133, T135)) → SPLITH_IN_GGAA(T134, T135)
The graph contains the following edges 3 >= 1, 2 >= 2, 4 > 2
PJ_IN_GGGAA(T133, T135, T134) → U23_GGGAA(T133, T135, T134, gtK_in_gg(T133, T135))
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(T164, T163, T167, T168, X216, X217, X218) → U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → PS_IN_GAGAGA(T167, X216, T168, X217, T163, X218)
PS_IN_GAGAGA(T167, T171, T168, X217, T163, X218) → U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → PT_IN_GAGGA(T168, X217, T171, T163, X218)
PT_IN_GAGGA(T168, T172, T171, T163, X218) → QSORTL_IN_GA(T168, T172)
QSORTL_IN_GA(.(T163, T164), X218) → PM_IN_GGAAAAA(T164, T163, X214, X215, X216, X217, X218)
PS_IN_GAGAGA(T167, T171, T168, X217, T163, X218) → QSORTL_IN_GA(T167, T171)

The TRS R consists of the following rules:

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(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(T164, T163, T167, T168, X216, X217, X218) → U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_GGAAAAA(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → PS_IN_GAGAGA(T167, X216, T168, X217, T163, X218)
PS_IN_GAGAGA(T167, T171, T168, X217, T163, X218) → U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_GAGAGA(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → PT_IN_GAGGA(T168, X217, T171, T163, X218)
PT_IN_GAGGA(T168, T172, T171, T163, X218) → QSORTL_IN_GA(T168, T172)
QSORTL_IN_GA(.(T163, T164), X218) → PM_IN_GGAAAAA(T164, T163, X214, X215, X216, X217, X218)
PS_IN_GAGAGA(T167, T171, T168, X217, T163, X218) → QSORTL_IN_GA(T167, T171)

The TRS R consists of the following rules:

splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
leG_in_gg(x1, x2) = leG_in_gg(x1, x2)
s(x1) = s(x1)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
0 = 0
leG_out_gg(x1, x2) = leG_out_gg(x1, x2)
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)
gtK_in_gg(x1, x2) = gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2) = gtK_out_gg(x1, x2)
U24_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)
qsortL_in_ga(x1, x2) = qsortL_in_ga(x1)
qsortL_out_ga(x1, x2) = qsortL_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)
QSORTL_IN_GA(x1, x2) = QSORTL_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(T164, T163) → U27_GGAAAAA(T164, T163, splitH_in_ggaa(T164, T163))
U27_GGAAAAA(T164, T163, splitH_out_ggaa(T164, T163, T167, T168)) → PS_IN_GAGAGA(T167, T168, T163)
PS_IN_GAGAGA(T167, T168, T163) → U29_GAGAGA(T167, T168, T163, qsortL_in_ga(T167))
U29_GAGAGA(T167, T168, T163, qsortL_out_ga(T167, T171)) → PT_IN_GAGGA(T168, T171, T163)
PT_IN_GAGGA(T168, T171, T163) → QSORTL_IN_GA(T168)
QSORTL_IN_GA(.(T163, T164)) → PM_IN_GGAAAAA(T164, T163)
PS_IN_GAGAGA(T167, T168, T163) → QSORTL_IN_GA(T167)

The TRS R consists of the following rules:

splitH_in_ggaa([], T105) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120) → U6_ggaa(T118, T119, T120, pI_in_gggaa(T118, T120, T119))
splitH_in_ggaa(.(T133, T134), T135) → U7_ggaa(T133, T134, T135, pJ_in_gggaa(T133, T135, T134))
qsortL_in_ga([]) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164)) → U9_ga(T163, T164, pM_in_ggaaaaa(T164, T163))
U6_ggaa(T118, T119, T120, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U7_ggaa(T133, T134, T135, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U9_ga(T163, T164, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
pI_in_gggaa(T118, T120, T119) → U21_gggaa(T118, T120, T119, leG_in_gg(T118, T120))
pJ_in_gggaa(T133, T135, T134) → U23_gggaa(T133, T135, T134, gtK_in_gg(T133, T135))
pM_in_ggaaaaa(T164, T163) → U27_ggaaaaa(T164, T163, splitH_in_ggaa(T164, T163))
U21_gggaa(T118, T120, T119, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, splitH_in_ggaa(T119, T120))
U23_gggaa(T133, T135, T134, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, splitH_in_ggaa(T134, T135))
U27_ggaaaaa(T164, T163, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, pS_in_gagaga(T167, T168, T163))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U22_gggaa(T118, T120, T119, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U24_gggaa(T133, T135, T134, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U28_ggaaaaa(T164, T163, T167, T168, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
pS_in_gagaga(T167, T168, T163) → U29_gagaga(T167, T168, T163, qsortL_in_ga(T167))
U29_gagaga(T167, T168, T163, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, T163, pT_in_gagga(T168, T171, T163))
U30_gagaga(T167, T171, T168, T163, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
pT_in_gagga(T168, T171, T163) → U31_gagga(T168, T171, T163, qsortL_in_ga(T168))
U31_gagga(T168, T171, T163, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, appendN_in_ggga(T171, T163, T172))
U32_gagga(T168, T172, T171, T163, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
appendN_in_ggga([], T185, T186) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198) → U10_ggga(T195, T196, T197, T198, appendN_in_ggga(T196, T197, T198))
U10_ggga(T195, T196, T197, T198, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))

The set Q consists of the following terms:

splitH_in_ggaa(x0, x1)
qsortL_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)
leG_in_gg(x0, x1)
U22_gggaa(x0, x1, x2, x3)
gtK_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.

U27_GGAAAAA(T164, T163, splitH_out_ggaa(T164, T163, T167, T168)) → PS_IN_GAGAGA(T167, T168, T163)

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

POL(.(x₁, x₂)) = 1 + x₂
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(QSORTL_IN_GA(x₁)) = x₁
POL(U10_ggga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U21_gggaa(x₁, x₂, x₃, x₄)) = x₃
POL(U22_gggaa(x₁, x₂, x₃, x₄)) = x₄
POL(U23_gggaa(x₁, x₂, x₃, x₄)) = x₃
POL(U24_gggaa(x₁, x₂, x₃, x₄)) = x₄
POL(U27_GGAAAAA(x₁, x₂, x₃)) = 1 + 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)) = 1 + 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(gtK_in_gg(x₁, x₂)) = 0
POL(gtK_out_gg(x₁, x₂)) = 0
POL(leG_in_gg(x₁, x₂)) = 0
POL(leG_out_gg(x₁, x₂)) = 0
POL(pI_in_gggaa(x₁, x₂, x₃)) = x₃
POL(pI_out_gggaa(x₁, x₂, x₃, x₄, x₅)) = 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(qsortL_in_ga(x₁)) = 0
POL(qsortL_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([], T105) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120) → U6_ggaa(T118, T119, T120, pI_in_gggaa(T118, T120, T119))
splitH_in_ggaa(.(T133, T134), T135) → U7_ggaa(T133, T134, T135, pJ_in_gggaa(T133, T135, T134))
U6_ggaa(T118, T119, T120, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
pI_in_gggaa(T118, T120, T119) → U21_gggaa(T118, T120, T119, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, splitH_in_ggaa(T119, T120))
U22_gggaa(T118, T120, T119, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U7_ggaa(T133, T134, T135, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
pJ_in_gggaa(T133, T135, T134) → U23_gggaa(T133, T135, T134, gtK_in_gg(T133, T135))
U23_gggaa(T133, T135, T134, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, splitH_in_ggaa(T134, T135))
U24_gggaa(T133, T135, T134, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)

(41) Obligation:

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

PM_IN_GGAAAAA(T164, T163) → U27_GGAAAAA(T164, T163, splitH_in_ggaa(T164, T163))
PS_IN_GAGAGA(T167, T168, T163) → U29_GAGAGA(T167, T168, T163, qsortL_in_ga(T167))
U29_GAGAGA(T167, T168, T163, qsortL_out_ga(T167, T171)) → PT_IN_GAGGA(T168, T171, T163)
PT_IN_GAGGA(T168, T171, T163) → QSORTL_IN_GA(T168)
QSORTL_IN_GA(.(T163, T164)) → PM_IN_GGAAAAA(T164, T163)
PS_IN_GAGAGA(T167, T168, T163) → QSORTL_IN_GA(T167)

The TRS R consists of the following rules:

splitH_in_ggaa([], T105) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120) → U6_ggaa(T118, T119, T120, pI_in_gggaa(T118, T120, T119))
splitH_in_ggaa(.(T133, T134), T135) → U7_ggaa(T133, T134, T135, pJ_in_gggaa(T133, T135, T134))
qsortL_in_ga([]) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164)) → U9_ga(T163, T164, pM_in_ggaaaaa(T164, T163))
U6_ggaa(T118, T119, T120, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U7_ggaa(T133, T134, T135, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U9_ga(T163, T164, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
pI_in_gggaa(T118, T120, T119) → U21_gggaa(T118, T120, T119, leG_in_gg(T118, T120))
pJ_in_gggaa(T133, T135, T134) → U23_gggaa(T133, T135, T134, gtK_in_gg(T133, T135))
pM_in_ggaaaaa(T164, T163) → U27_ggaaaaa(T164, T163, splitH_in_ggaa(T164, T163))
U21_gggaa(T118, T120, T119, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, splitH_in_ggaa(T119, T120))
U23_gggaa(T133, T135, T134, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, splitH_in_ggaa(T134, T135))
U27_ggaaaaa(T164, T163, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, pS_in_gagaga(T167, T168, T163))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U22_gggaa(T118, T120, T119, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U24_gggaa(T133, T135, T134, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U28_ggaaaaa(T164, T163, T167, T168, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
pS_in_gagaga(T167, T168, T163) → U29_gagaga(T167, T168, T163, qsortL_in_ga(T167))
U29_gagaga(T167, T168, T163, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, T163, pT_in_gagga(T168, T171, T163))
U30_gagaga(T167, T171, T168, T163, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
pT_in_gagga(T168, T171, T163) → U31_gagga(T168, T171, T163, qsortL_in_ga(T168))
U31_gagga(T168, T171, T163, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, appendN_in_ggga(T171, T163, T172))
U32_gagga(T168, T172, T171, T163, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
appendN_in_ggga([], T185, T186) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198) → U10_ggga(T195, T196, T197, T198, appendN_in_ggga(T196, T197, T198))
U10_ggga(T195, T196, T197, T198, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))

The set Q consists of the following terms:

splitH_in_ggaa(x0, x1)
qsortL_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)
leG_in_gg(x0, x1)
U22_gggaa(x0, x1, x2, x3)
gtK_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(.(T49, T50), T51, T52, .(T49, T54)) → APPENDF_IN_GGGA(T50, T51, T52, T54)

The TRS R consists of the following rules:

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(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(.(T49, T50), T51, T52, .(T49, T54)) → APPENDF_IN_GGGA(T50, T51, T52, T54)

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

QSORTA_IN_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, leG_in_gg(T71, T73))
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, leG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, X84, X85, T71, X12, X13, T9)
PP_IN_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9) → U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → PQ_IN_GGAGAGA(T71, T97, X12, T98, X13, T73, T9)
PQ_IN_GGAGAGA(T71, T97, T157, T98, X13, T73, T9) → QSORTA_IN_GA(.(T71, T97), T157)

The TRS R consists of the following rules:

qsortA_in_ga([], []) → qsortA_out_ga([], [])
qsortA_in_ga(.(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, qsortE_in_a(T19))
qsortE_in_a([]) → qsortE_out_a([])
U11_aaga(T19, X13, T14, T9, qsortE_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, qsortE_in_a(T24))
U13_agga(T24, T19, T14, T9, qsortE_out_a(T24)) → U14_agga(T24, T19, T14, T9, appendF_in_ggga(T19, T14, T24, T9))
appendF_in_ggga([], T37, T38, .(T37, T38)) → appendF_out_ggga([], T37, T38, .(T37, T38))
appendF_in_ggga(.(T49, T50), T51, T52, .(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(.(T49, T50), T51, T52, .(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)) → qsortA_out_ga(.(T14, []), T9)
qsortA_in_ga(.(T73, .(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, leG_in_gg(T71, T73))
leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U15_gggaaaaa(T71, T73, T72, X84, X85, X12, X13, T9, leG_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, T97, T98, T71, X12, X13, T9) → U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U17_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_in_ggagaga(T71, T97, X12, T98, X13, T73, T9))
pQ_in_ggagaga(T71, T97, T157, T98, X13, T73, T9) → U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_in_ga(.(T71, T97), T157))
qsortA_in_ga(.(T215, .(T213, T214)), T9) → U3_ga(T215, T213, T214, T9, pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9))
pD_in_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9) → U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_in_gg(T213, T215))
U33_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, gtK_out_gg(T213, T215)) → U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_in_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9))
pU_in_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9) → U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_in_ggaa(T214, T215, T222, T223))
U35_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, splitH_out_ggaa(T214, T215, T222, T223)) → U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_in_gaggaga(T222, X12, T213, T223, X13, T215, T9))
pV_in_gaggaga(T222, T226, T213, T223, X13, T215, T9) → U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_in_ga(T222, T226))
qsortL_in_ga([], []) → qsortL_out_ga([], [])
qsortL_in_ga(.(T163, T164), X218) → U9_ga(T163, T164, X218, pM_in_ggaaaaa(T164, T163, X214, X215, X216, X217, X218))
pM_in_ggaaaaa(T164, T163, T167, T168, X216, X217, X218) → U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_in_ggaa(T164, T163, T167, T168))
U27_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, splitH_out_ggaa(T164, T163, T167, T168)) → U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_in_gagaga(T167, X216, T168, X217, T163, X218))
pS_in_gagaga(T167, T171, T168, X217, T163, X218) → U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_in_ga(T167, T171))
U29_gagaga(T167, T171, T168, X217, T163, X218, qsortL_out_ga(T167, T171)) → U30_gagaga(T167, T171, T168, X217, T163, X218, pT_in_gagga(T168, X217, T171, T163, X218))
pT_in_gagga(T168, T172, T171, T163, X218) → U31_gagga(T168, T172, T171, T163, X218, qsortL_in_ga(T168, T172))
U31_gagga(T168, T172, T171, T163, X218, qsortL_out_ga(T168, T172)) → U32_gagga(T168, T172, T171, T163, X218, appendN_in_ggga(T171, T163, T172, X218))
appendN_in_ggga([], T185, T186, .(T185, T186)) → appendN_out_ggga([], T185, T186, .(T185, T186))
appendN_in_ggga(.(T195, T196), T197, T198, .(T195, X244)) → U10_ggga(T195, T196, T197, T198, X244, appendN_in_ggga(T196, T197, T198, X244))
U10_ggga(T195, T196, T197, T198, X244, appendN_out_ggga(T196, T197, T198, X244)) → appendN_out_ggga(.(T195, T196), T197, T198, .(T195, X244))
U32_gagga(T168, T172, T171, T163, X218, appendN_out_ggga(T171, T163, T172, X218)) → pT_out_gagga(T168, T172, T171, T163, X218)
U30_gagaga(T167, T171, T168, X217, T163, X218, pT_out_gagga(T168, X217, T171, T163, X218)) → pS_out_gagaga(T167, T171, T168, X217, T163, X218)
U28_ggaaaaa(T164, T163, T167, T168, X216, X217, X218, pS_out_gagaga(T167, X216, T168, X217, T163, X218)) → pM_out_ggaaaaa(T164, T163, T167, T168, X216, X217, X218)
U9_ga(T163, T164, X218, pM_out_ggaaaaa(T164, T163, X214, X215, X216, X217, X218)) → qsortL_out_ga(.(T163, T164), X218)
U37_gaggaga(T222, T226, T213, T223, X13, T215, T9, qsortL_out_ga(T222, T226)) → U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_in_gagga(.(T213, T223), X13, T226, T215, T9))
pR_in_gagga(T98, T158, T157, T73, T9) → U25_gagga(T98, T158, T157, T73, T9, qsortL_in_ga(T98, T158))
U25_gagga(T98, T158, T157, T73, T9, qsortL_out_ga(T98, T158)) → U26_gagga(T98, T158, T157, T73, T9, appendF_in_ggga(T157, T73, T158, T9))
U26_gagga(T98, T158, T157, T73, T9, appendF_out_ggga(T157, T73, T158, T9)) → pR_out_gagga(T98, T158, T157, T73, T9)
U38_gaggaga(T222, T226, T213, T223, X13, T215, T9, pR_out_gagga(.(T213, T223), X13, T226, T215, T9)) → pV_out_gaggaga(T222, T226, T213, T223, X13, T215, T9)
U36_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9, pV_out_gaggaga(T222, X12, T213, T223, X13, T215, T9)) → pU_out_ggaaagaa(T214, T215, T222, T223, X12, T213, X13, T9)
U34_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9, pU_out_ggaaagaa(T214, T215, X268, X269, X12, T213, X13, T9)) → pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)
U3_ga(T215, T213, T214, T9, pD_out_gggaaaaa(T213, T215, T214, X268, X269, X12, X13, T9)) → qsortA_out_ga(.(T215, .(T213, T214)), T9)
U19_ggagaga(T71, T97, T157, T98, X13, T73, T9, qsortA_out_ga(.(T71, T97), T157)) → U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_in_gagga(T98, X13, T157, T73, T9))
U20_ggagaga(T71, T97, T157, T98, X13, T73, T9, pR_out_gagga(T98, X13, T157, T73, T9)) → pQ_out_ggagaga(T71, T97, T157, T98, X13, T73, T9)
U18_ggaagaaa(T72, T73, T97, T98, T71, X12, X13, T9, pQ_out_ggagaga(T71, T97, X12, T98, X13, T73, T9)) → pP_out_ggaagaaa(T72, T73, T97, T98, 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)) → qsortA_out_ga(.(T73, .(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:

QSORTA_IN_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, leG_in_gg(T71, T73))
U15_GGGAAAAA(T71, T73, T72, X84, X85, X12, X13, T9, leG_out_gg(T71, T73)) → PP_IN_GGAAGAAA(T72, T73, X84, X85, T71, X12, X13, T9)
PP_IN_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9) → U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_in_ggaa(T72, T73, T97, T98))
U17_GGAAGAAA(T72, T73, T97, T98, T71, X12, X13, T9, splitH_out_ggaa(T72, T73, T97, T98)) → PQ_IN_GGAGAGA(T71, T97, X12, T98, X13, T73, T9)
PQ_IN_GGAGAGA(T71, T97, T157, T98, X13, T73, T9) → QSORTA_IN_GA(.(T71, T97), T157)

The TRS R consists of the following rules:

leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
splitH_in_ggaa([], T105, [], []) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120, .(T118, X149), X150) → U6_ggaa(T118, T119, T120, X149, X150, pI_in_gggaa(T118, T120, T119, X149, X150))
splitH_in_ggaa(.(T133, T134), T135, X176, .(T133, X177)) → U7_ggaa(T133, T134, T135, X176, X177, pJ_in_gggaa(T133, T135, T134, X176, X177))
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U6_ggaa(T118, T119, T120, X149, X150, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U7_ggaa(T133, T134, T135, X176, X177, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
pI_in_gggaa(T118, T120, T119, X149, X150) → U21_gggaa(T118, T120, T119, X149, X150, leG_in_gg(T118, T120))
pJ_in_gggaa(T133, T135, T134, X176, X177) → U23_gggaa(T133, T135, T134, X176, X177, gtK_in_gg(T133, T135))
U21_gggaa(T118, T120, T119, X149, X150, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, X149, X150, splitH_in_ggaa(T119, T120, X149, X150))
U23_gggaa(T133, T135, T134, X176, X177, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, X176, X177, splitH_in_ggaa(T134, T135, X176, X177))
U22_gggaa(T118, T120, T119, X149, X150, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U24_gggaa(T133, T135, T134, X176, X177, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))

(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:

QSORTA_IN_GA(.(T73, .(T71, T72))) → PC_IN_GGGAAAAA(T71, T73, T72)
PC_IN_GGGAAAAA(T71, T73, T72) → U15_GGGAAAAA(T71, T73, T72, leG_in_gg(T71, T73))
U15_GGGAAAAA(T71, T73, T72, leG_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, T97, T98)) → PQ_IN_GGAGAGA(T71, T97, T98, T73)
PQ_IN_GGAGAGA(T71, T97, T98, T73) → QSORTA_IN_GA(.(T71, T97))

The TRS R consists of the following rules:

leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
splitH_in_ggaa([], T105) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120) → U6_ggaa(T118, T119, T120, pI_in_gggaa(T118, T120, T119))
splitH_in_ggaa(.(T133, T134), T135) → U7_ggaa(T133, T134, T135, pJ_in_gggaa(T133, T135, T134))
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U6_ggaa(T118, T119, T120, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U7_ggaa(T133, T134, T135, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
pI_in_gggaa(T118, T120, T119) → U21_gggaa(T118, T120, T119, leG_in_gg(T118, T120))
pJ_in_gggaa(T133, T135, T134) → U23_gggaa(T133, T135, T134, gtK_in_gg(T133, T135))
U21_gggaa(T118, T120, T119, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, splitH_in_ggaa(T119, T120))
U23_gggaa(T133, T135, T134, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, splitH_in_ggaa(T134, T135))
U22_gggaa(T118, T120, T119, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U24_gggaa(T133, T135, T134, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))

The set Q consists of the following terms:

leG_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)
gtK_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.

QSORTA_IN_GA(.(T73, .(T71, T72))) → PC_IN_GGGAAAAA(T71, T73, T72)

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

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(PC_IN_GGGAAAAA(x₁, x₂, x₃)) = 1 + x₃
POL(PP_IN_GGAAGAAA(x₁, x₂, x₃)) = 1 + x₁
POL(PQ_IN_GGAGAGA(x₁, x₂, x₃, x₄)) = 1 + x₂
POL(QSORTA_IN_GA(x₁)) = x₁
POL(U15_GGGAAAAA(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U17_GGAAGAAA(x₁, x₂, x₃, x₄)) = x₄
POL(U21_gggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U22_gggaa(x₁, x₂, x₃, x₄)) = x₄
POL(U23_gggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U24_gggaa(x₁, x₂, x₃, x₄)) = x₄
POL(U5_gg(x₁, x₂, x₃)) = 0
POL(U6_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(U7_ggaa(x₁, x₂, x₃, x₄)) = x₄
POL(U8_gg(x₁, x₂, x₃)) = 0
POL([]) = 0
POL(gtK_in_gg(x₁, x₂)) = 0
POL(gtK_out_gg(x₁, x₂)) = 0
POL(leG_in_gg(x₁, x₂)) = 0
POL(leG_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
POL(pJ_in_gggaa(x₁, x₂, x₃)) = 1 + x₃
POL(pJ_out_gggaa(x₁, x₂, x₃, x₄, x₅)) = 1 + x₄
POL(s(x₁)) = 0
POL(splitH_in_ggaa(x₁, x₂)) = 1 + x₁
POL(splitH_out_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃

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

splitH_in_ggaa([], T105) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120) → U6_ggaa(T118, T119, T120, pI_in_gggaa(T118, T120, T119))
splitH_in_ggaa(.(T133, T134), T135) → U7_ggaa(T133, T134, T135, pJ_in_gggaa(T133, T135, T134))
U21_gggaa(T118, T120, T119, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, splitH_in_ggaa(T119, T120))
U22_gggaa(T118, T120, T119, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
U6_ggaa(T118, T119, T120, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
pI_in_gggaa(T118, T120, T119) → U21_gggaa(T118, T120, T119, leG_in_gg(T118, T120))
U7_ggaa(T133, T134, T135, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
pJ_in_gggaa(T133, T135, T134) → U23_gggaa(T133, T135, T134, gtK_in_gg(T133, T135))
U23_gggaa(T133, T135, T134, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, splitH_in_ggaa(T134, T135))
U24_gggaa(T133, T135, T134, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)

(57) Obligation:

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

PC_IN_GGGAAAAA(T71, T73, T72) → U15_GGGAAAAA(T71, T73, T72, leG_in_gg(T71, T73))
U15_GGGAAAAA(T71, T73, T72, leG_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, T97, T98)) → PQ_IN_GGAGAGA(T71, T97, T98, T73)
PQ_IN_GGAGAGA(T71, T97, T98, T73) → QSORTA_IN_GA(.(T71, T97))

The TRS R consists of the following rules:

leG_in_gg(s(T86), s(T87)) → U5_gg(T86, T87, leG_in_gg(T86, T87))
leG_in_gg(0, s(T94)) → leG_out_gg(0, s(T94))
leG_in_gg(0, 0) → leG_out_gg(0, 0)
splitH_in_ggaa([], T105) → splitH_out_ggaa([], T105, [], [])
splitH_in_ggaa(.(T118, T119), T120) → U6_ggaa(T118, T119, T120, pI_in_gggaa(T118, T120, T119))
splitH_in_ggaa(.(T133, T134), T135) → U7_ggaa(T133, T134, T135, pJ_in_gggaa(T133, T135, T134))
U5_gg(T86, T87, leG_out_gg(T86, T87)) → leG_out_gg(s(T86), s(T87))
U6_ggaa(T118, T119, T120, pI_out_gggaa(T118, T120, T119, X149, X150)) → splitH_out_ggaa(.(T118, T119), T120, .(T118, X149), X150)
U7_ggaa(T133, T134, T135, pJ_out_gggaa(T133, T135, T134, X176, X177)) → splitH_out_ggaa(.(T133, T134), T135, X176, .(T133, X177))
pI_in_gggaa(T118, T120, T119) → U21_gggaa(T118, T120, T119, leG_in_gg(T118, T120))
pJ_in_gggaa(T133, T135, T134) → U23_gggaa(T133, T135, T134, gtK_in_gg(T133, T135))
U21_gggaa(T118, T120, T119, leG_out_gg(T118, T120)) → U22_gggaa(T118, T120, T119, splitH_in_ggaa(T119, T120))
U23_gggaa(T133, T135, T134, gtK_out_gg(T133, T135)) → U24_gggaa(T133, T135, T134, splitH_in_ggaa(T134, T135))
U22_gggaa(T118, T120, T119, splitH_out_ggaa(T119, T120, X149, X150)) → pI_out_gggaa(T118, T120, T119, X149, X150)
gtK_in_gg(s(T148), s(T149)) → U8_gg(T148, T149, gtK_in_gg(T148, T149))
gtK_in_gg(s(T154), 0) → gtK_out_gg(s(T154), 0)
U24_gggaa(T133, T135, T134, splitH_out_ggaa(T134, T135, X176, X177)) → pJ_out_gggaa(T133, T135, T134, X176, X177)
U8_gg(T148, T149, gtK_out_gg(T148, T149)) → gtK_out_gg(s(T148), s(T149))

The set Q consists of the following terms:

leG_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)
gtK_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