Left Termination of the query pattern qsort_in_2(g, a) w.r.t. the given Prolog program could not be shown:

(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).

Queries:

qsort(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: qsort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: qsort(T1, T2), SCOPE: 1, CLAUSE: 0 | qsort(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | qsort([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: qsort(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nqsort(T1, T2) !=? qsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: qsort([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(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];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(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];
10 [label="10: ','(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];
11 [label="11: ','(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];
12 [label="12: ','(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];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: qsort([], X12)\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(qsort([], X13), append(T17, .(T14, X13), T9))\n\nT14 is ground\nT17 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: qsort([], X12), SCOPE: 3, CLAUSE: 0 | qsort([], X12), SCOPE: 3, CLAUSE: 1\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: qsort([], X12), SCOPE: 3, CLAUSE: 0\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: qsort([], X12), SCOPE: 3, CLAUSE: 1\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: qsort([], X13)\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: append(T17, .(T14, T20), T9)\n\nT14 is ground\nT17 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: append(T17, .(T14, T20), T9), SCOPE: 4, CLAUSE: 5 | append(T17, .(T14, T20), T9), SCOPE: 4, CLAUSE: 6\n\nT14 is ground\nT17 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: append(T17, .(T14, T20), T9), SCOPE: 4, CLAUSE: 5\n\nT14 is ground\nT17 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: append(T17, .(T14, T20), T9), SCOPE: 4, CLAUSE: 6\n\nT14 is ground\nT17 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: append(T46, .(T47, T48), T50)\n\nT46 is ground\nT47 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(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];
33 [label="33: ','(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="34: ','(le(T67, T69), ','(split(T68, T69, X93, X94), ','(qsort(.(T67, X93), X12), ','(qsort(X94, X13), append(X12, .(T69, X13), T9)))))\n\nT67 is ground\nT68 is ground\nT69 is ground\nX12 is free; \nX13 is free; \nX93 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: le(T67, T69)\n\nT67 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(split(T68, T69, X93, X94), ','(qsort(.(T67, X93), X12), ','(qsort(X94, X13), append(X12, .(T69, X13), T9))))\n\nT67 is ground\nT68 is ground\nT69 is ground\nX12 is free; \nX13 is free; \nX93 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: le(T67, T69), SCOPE: 5, CLAUSE: 9 | le(T67, T69), SCOPE: 5, CLAUSE: 10 | le(T67, T69), SCOPE: 5, CLAUSE: 11\n\nT67 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: le(T67, T69), SCOPE: 5, CLAUSE: 9\n\nT67 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: le(T67, T69), SCOPE: 5, CLAUSE: 10 | le(T67, T69), SCOPE: 5, CLAUSE: 11\n\nT67 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: le(T82, T83)\n\nT82 is ground\nT83 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: le(T67, T69), SCOPE: 5, CLAUSE: 10\n\nT67 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: le(T67, T69), SCOPE: 5, CLAUSE: 11\n\nT67 is ground\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: split(T68, T69, X93, X94)\n\nT68 is ground\nT69 is ground\nX93 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(qsort(.(T67, T93), X12), ','(qsort(T94, X13), append(X12, .(T69, X13), T9)))\n\nT67 is ground\nT69 is ground\nT93 is ground\nT94 is ground\nX12 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: split(T68, T69, X93, X94), SCOPE: 6, CLAUSE: 2 | split(T68, T69, X93, X94), SCOPE: 6, CLAUSE: 3 | split(T68, T69, X93, X94), SCOPE: 6, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nX93 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: split(T68, T69, X93, X94), SCOPE: 6, CLAUSE: 2\n\nT68 is ground\nT69 is ground\nX93 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: split(T68, T69, X93, X94), SCOPE: 6, CLAUSE: 3 | split(T68, T69, X93, X94), SCOPE: 6, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nX93 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: split(T68, T69, X93, X94), SCOPE: 6, CLAUSE: 3\n\nT68 is ground\nT69 is ground\nX93 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: split(T68, T69, X93, X94), SCOPE: 6, CLAUSE: 4\n\nT68 is ground\nT69 is ground\nX93 is free; \nX94 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(le(T114, T116), split(T115, T116, X167, X168))\n\nT114 is ground\nT115 is ground\nT116 is ground\nX167 is free; \nX168 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: le(T114, T116)\n\nT114 is ground\nT116 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: split(T115, T116, X167, X168)\n\nT115 is ground\nT116 is ground\nX167 is free; \nX168 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(gt(T129, T131), split(T130, T131, X199, X200))\n\nT129 is ground\nT130 is ground\nT131 is ground\nX199 is free; \nX200 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: gt(T129, T131)\n\nT129 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: split(T130, T131, X199, X200)\n\nT130 is ground\nT131 is ground\nX199 is free; \nX200 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: gt(T129, T131), SCOPE: 7, CLAUSE: 7 | gt(T129, T131), SCOPE: 7, CLAUSE: 8\n\nT129 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: gt(T129, T131), SCOPE: 7, CLAUSE: 7\n\nT129 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: gt(T129, T131), SCOPE: 7, CLAUSE: 8\n\nT129 is ground\nT131 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: gt(T144, T145)\n\nT144 is ground\nT145 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: qsort(.(T67, T93), X12)\n\nT67 is ground\nT93 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(qsort(T94, X13), append(T155, .(T69, X13), T9))\n\nT69 is ground\nT94 is ground\nT155 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: qsort(T94, X13)\n\nT94 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: append(T155, .(T69, T158), T9)\n\nT69 is ground\nT155 is ground\nT158 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: qsort(T94, X13), SCOPE: 8, CLAUSE: 0 | qsort(T94, X13), SCOPE: 8, CLAUSE: 1\n\nT94 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: qsort(T94, X13), SCOPE: 8, CLAUSE: 0\n\nT94 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: qsort(T94, X13), SCOPE: 8, CLAUSE: 1\n\nT94 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(split(T164, T163, X261, X262), ','(qsort(X261, X263), ','(qsort(X262, X264), append(X263, .(T163, X264), X265))))\n\nT163 is ground\nT164 is ground\nX265 is free; \nX261 is free; \nX262 is free; \nX263 is free; \nX264 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: split(T164, T163, X261, X262)\n\nT163 is ground\nT164 is ground\nX261 is free; \nX262 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(qsort(T167, X263), ','(qsort(T168, X264), append(X263, .(T163, X264), X265)))\n\nT163 is ground\nT167 is ground\nT168 is ground\nX265 is free; \nX263 is free; \nX264 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: qsort(T167, X263)\n\nT167 is ground\nX263 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(qsort(T168, X264), append(T171, .(T163, X264), X265))\n\nT163 is ground\nT168 is ground\nT171 is ground\nX265 is free; \nX264 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: qsort(T168, X264)\n\nT168 is ground\nX264 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: append(T171, .(T163, T172), X265)\n\nT163 is ground\nT171 is ground\nT172 is ground\nX265 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: append(T171, .(T163, T172), X265), SCOPE: 9, CLAUSE: 5 | append(T171, .(T163, T172), X265), SCOPE: 9, CLAUSE: 6\n\nT163 is ground\nT171 is ground\nT172 is ground\nX265 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: append(T171, .(T163, T172), X265), SCOPE: 9, CLAUSE: 5\n\nT163 is ground\nT171 is ground\nT172 is ground\nX265 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: append(T171, .(T163, T172), X265), SCOPE: 9, CLAUSE: 6\n\nT163 is ground\nT171 is ground\nT172 is ground\nX265 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: append(T196, .(T197, T198), X295)\n\nT196 is ground\nT197 is ground\nT198 is ground\nX295 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(gt(T213, T215), ','(split(T214, T215, X323, X324), ','(qsort(X323, X12), ','(qsort(.(T213, X324), X13), append(X12, .(T215, X13), T9)))))\n\nT213 is ground\nT214 is ground\nT215 is ground\nX12 is free; \nX13 is free; \nX323 is free; \nX324 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: gt(T213, T215)\n\nT213 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: ','(split(T214, T215, X323, X324), ','(qsort(X323, X12), ','(qsort(.(T213, X324), X13), append(X12, .(T215, X13), T9))))\n\nT213 is ground\nT214 is ground\nT215 is ground\nX12 is free; \nX13 is free; \nX323 is free; \nX324 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: split(T214, T215, X323, X324)\n\nT214 is ground\nT215 is ground\nX323 is free; \nX324 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: ','(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];
109 [label="109: qsort(T222, X12)\n\nT222 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: ','(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];
111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 111 [arrowhead = none ];
111 -> 2;

112 [label="EVAL with clause\nqsort([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 112 [arrowhead = none ];
112 -> 3;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 113 [arrowhead = none ];
113 -> 4;

114 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 114 [arrowhead = none ];
114 -> 5;

115 [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 substitution\nX7 -> T6\nX8 -> T7\nT1 -> .(T6, T7)\nT2 -> T9\nX9 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 115 [arrowhead = none ];
115 -> 7;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 116 [arrowhead = none ];
116 -> 8;

117 [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 = white];
5 -> 117 [arrowhead = none ];
117 -> 6;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 118 [arrowhead = none ];
118 -> 9;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 119 [arrowhead = none ];
119 -> 10;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 120 [arrowhead = none ];
120 -> 11;

121 [label="EVAL with clause\nsplit([], X21, [], []).\nand substitution\nT7 -> []\nT6 -> T14\nX21 -> T14\nX10 -> []\nX11 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 121 [arrowhead = none ];
121 -> 12;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 122 [arrowhead = none ];
122 -> 13;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 123 [arrowhead = none ];
123 -> 32;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 124 [arrowhead = none ];
124 -> 33;

125 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
12 -> 125 [arrowhead = none ];
125 -> 14;

126 [label="SPLIT 2\nnew knowledge:\nT17 is ground\nreplacements:\nX12 -> T17", fontsize=14, style = filled, fillcolor = violet];
12 -> 126 [arrowhead = none ];
126 -> 15;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 127 [arrowhead = none ];
127 -> 16;

128 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 128 [arrowhead = none ];
128 -> 22;

129 [label="SPLIT 2\nnew knowledge:\nT20 is ground\nreplacements:\nX13 -> T20", fontsize=14, style = filled, fillcolor = violet];
15 -> 129 [arrowhead = none ];
129 -> 23;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 130 [arrowhead = none ];
130 -> 17;

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 131 [arrowhead = none ];
131 -> 18;

132 [label="ONLY EVAL with clause\nqsort([], []).\nand substitution\nX12 -> []", fontsize=14, style = filled, fillcolor = white];
17 -> 132 [arrowhead = none ];
132 -> 19;

133 [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 = white];
18 -> 133 [arrowhead = none ];
133 -> 21;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 134 [arrowhead = none ];
134 -> 20;

135 [label="INSTANCE with matching:\nX12 -> X13", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 135 [arrowhead = none , style=dashed];
135 -> 14[style=dashed];

136 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 136 [arrowhead = none ];
136 -> 24;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 137 [arrowhead = none ];
137 -> 25;

138 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 138 [arrowhead = none ];
138 -> 26;

139 [label="EVAL with clause\nappend([], X40, X40).\nand substitution\nT17 -> []\nT14 -> T33\nT20 -> T34\nX40 -> .(T33, T34)\nT9 -> .(T33, T34)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 139 [arrowhead = none ];
139 -> 27;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 140 [arrowhead = none ];
140 -> 28;

141 [label="EVAL with clause\nappend(.(X50, X51), X52, .(X50, X53)) :- append(X51, X52, X53).\nand substitution\nX50 -> T45\nX51 -> T46\nT17 -> .(T45, T46)\nT14 -> T47\nT20 -> T48\nX52 -> .(T47, T48)\nX53 -> T50\nT9 -> .(T45, T50)\nT49 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 141 [arrowhead = none ];
141 -> 30;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 142 [arrowhead = none ];
142 -> 31;

143 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 143 [arrowhead = none ];
143 -> 29;

144 [label="INSTANCE with matching:\nT17 -> T46\nT14 -> T47\nT20 -> T48\nT9 -> T50", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 144 [arrowhead = none , style=dashed];
144 -> 23[style=dashed];

145 [label="EVAL with clause\nsplit(.(X88, X89), X90, .(X88, X91), X92) :- ','(le(X88, X90), split(X89, X90, X91, X92)).\nand substitution\nX88 -> T67\nX89 -> T68\nT7 -> .(T67, T68)\nT6 -> T69\nX90 -> T69\nX91 -> X93\nX10 -> .(T67, X93)\nX11 -> X94\nX92 -> X94", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 145 [arrowhead = none ];
145 -> 34;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 146 [arrowhead = none ];
146 -> 35;

147 [label="EVAL with clause\nsplit(.(X318, X319), X320, X321, .(X318, X322)) :- ','(gt(X318, X320), split(X319, X320, X321, X322)).\nand substitution\nX318 -> T213\nX319 -> T214\nT7 -> .(T213, T214)\nT6 -> T215\nX320 -> T215\nX10 -> X323\nX321 -> X323\nX322 -> X324\nX11 -> .(T213, X324)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 147 [arrowhead = none ];
147 -> 103;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 148 [arrowhead = none ];
148 -> 104;

149 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
34 -> 149 [arrowhead = none ];
149 -> 36;

150 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
34 -> 150 [arrowhead = none ];
150 -> 37;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 151 [arrowhead = none ];
151 -> 38;

152 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
37 -> 152 [arrowhead = none ];
152 -> 51;

153 [label="SPLIT 2\nnew knowledge:\nT93 is ground\nT94 is ground\nreplacements:\nX93 -> T93\nX94 -> T94", fontsize=14, style = filled, fillcolor = violet];
37 -> 153 [arrowhead = none ];
153 -> 52;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 154 [arrowhead = none ];
154 -> 39;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 155 [arrowhead = none ];
155 -> 40;

156 [label="EVAL with clause\nle(s(X112), s(X113)) :- le(X112, X113).\nand substitution\nX112 -> T82\nT67 -> s(T82)\nX113 -> T83\nT69 -> s(T83)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 156 [arrowhead = none ];
156 -> 41;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 157 [arrowhead = none ];
157 -> 42;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 158 [arrowhead = none ];
158 -> 43;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 159 [arrowhead = none ];
159 -> 44;

160 [label="INSTANCE with matching:\nT67 -> T82\nT69 -> T83", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 160 [arrowhead = none , style=dashed];
160 -> 36[style=dashed];

161 [label="EVAL with clause\nle(0, s(X122)).\nand substitution\nT67 -> 0\nX122 -> T90\nT69 -> s(T90)", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 161 [arrowhead = none ];
161 -> 45;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 162 [arrowhead = none ];
162 -> 46;

163 [label="EVAL with clause\nle(0, 0).\nand substitution\nT67 -> 0\nT69 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 163 [arrowhead = none ];
163 -> 48;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 164 [arrowhead = none ];
164 -> 49;

165 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 165 [arrowhead = none ];
165 -> 47;

166 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 166 [arrowhead = none ];
166 -> 50;

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 167 [arrowhead = none ];
167 -> 53;

168 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 168 [arrowhead = none ];
168 -> 77;

169 [label="SPLIT 2\nnew knowledge:\nT155 is ground\nreplacements:\nX12 -> T155", fontsize=14, style = filled, fillcolor = violet];
52 -> 169 [arrowhead = none ];
169 -> 78;

170 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 170 [arrowhead = none ];
170 -> 54;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 171 [arrowhead = none ];
171 -> 55;

172 [label="EVAL with clause\nsplit([], X132, [], []).\nand substitution\nT68 -> []\nT69 -> T101\nX132 -> T101\nX93 -> []\nX94 -> []", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 172 [arrowhead = none ];
172 -> 56;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 173 [arrowhead = none ];
173 -> 57;

174 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 174 [arrowhead = none ];
174 -> 59;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 175 [arrowhead = none ];
175 -> 60;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 176 [arrowhead = none ];
176 -> 58;

177 [label="EVAL with clause\nsplit(.(X162, X163), X164, .(X162, X165), X166) :- ','(le(X162, X164), split(X163, X164, X165, X166)).\nand substitution\nX162 -> T114\nX163 -> T115\nT68 -> .(T114, T115)\nT69 -> T116\nX164 -> T116\nX165 -> X167\nX93 -> .(T114, X167)\nX94 -> X168\nX166 -> X168", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 177 [arrowhead = none ];
177 -> 61;

178 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 178 [arrowhead = none ];
178 -> 62;

179 [label="EVAL with clause\nsplit(.(X194, X195), X196, X197, .(X194, X198)) :- ','(gt(X194, X196), split(X195, X196, X197, X198)).\nand substitution\nX194 -> T129\nX195 -> T130\nT68 -> .(T129, T130)\nT69 -> T131\nX196 -> T131\nX93 -> X199\nX197 -> X199\nX198 -> X200\nX94 -> .(T129, X200)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 179 [arrowhead = none ];
179 -> 65;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 180 [arrowhead = none ];
180 -> 66;

181 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 181 [arrowhead = none ];
181 -> 63;

182 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
61 -> 182 [arrowhead = none ];
182 -> 64;

183 [label="INSTANCE with matching:\nT67 -> T114\nT69 -> T116", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 183 [arrowhead = none , style=dashed];
183 -> 36[style=dashed];

184 [label="INSTANCE with matching:\nT68 -> T115\nT69 -> T116\nX93 -> X167\nX94 -> X168", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 184 [arrowhead = none , style=dashed];
184 -> 51[style=dashed];

185 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
65 -> 185 [arrowhead = none ];
185 -> 67;

186 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
65 -> 186 [arrowhead = none ];
186 -> 68;

187 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 187 [arrowhead = none ];
187 -> 69;

188 [label="INSTANCE with matching:\nT68 -> T130\nT69 -> T131\nX93 -> X199\nX94 -> X200", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 188 [arrowhead = none , style=dashed];
188 -> 51[style=dashed];

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 189 [arrowhead = none ];
189 -> 70;

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 190 [arrowhead = none ];
190 -> 71;

191 [label="EVAL with clause\ngt(s(X218), s(X219)) :- gt(X218, X219).\nand substitution\nX218 -> T144\nT129 -> s(T144)\nX219 -> T145\nT131 -> s(T145)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 191 [arrowhead = none ];
191 -> 72;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 192 [arrowhead = none ];
192 -> 73;

193 [label="EVAL with clause\ngt(s(X226), 0).\nand substitution\nX226 -> T150\nT129 -> s(T150)\nT131 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 193 [arrowhead = none ];
193 -> 74;

194 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 194 [arrowhead = none ];
194 -> 75;

195 [label="INSTANCE with matching:\nT129 -> T144\nT131 -> T145", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 195 [arrowhead = none , style=dashed];
195 -> 67[style=dashed];

196 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 196 [arrowhead = none ];
196 -> 76;

197 [label="INSTANCE with matching:\nT1 -> .(T67, T93)\nT2 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 197 [arrowhead = none , style=dashed];
197 -> 1[style=dashed];

198 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
78 -> 198 [arrowhead = none ];
198 -> 79;

199 [label="SPLIT 2\nnew knowledge:\nT158 is ground\nreplacements:\nX13 -> T158", fontsize=14, style = filled, fillcolor = violet];
78 -> 199 [arrowhead = none ];
199 -> 80;

200 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
79 -> 200 [arrowhead = none ];
200 -> 81;

201 [label="INSTANCE with matching:\nT17 -> T155\nT14 -> T69\nT20 -> T158", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 201 [arrowhead = none , style=dashed];
201 -> 23[style=dashed];

202 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 202 [arrowhead = none ];
202 -> 82;

203 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 203 [arrowhead = none ];
203 -> 83;

204 [label="EVAL with clause\nqsort([], []).\nand substitution\nT94 -> []\nX13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 204 [arrowhead = none ];
204 -> 84;

205 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 205 [arrowhead = none ];
205 -> 85;

206 [label="EVAL with clause\nqsort(.(X258, X259), X260) :- ','(split(X259, X258, X261, X262), ','(qsort(X261, X263), ','(qsort(X262, X264), append(X263, .(X258, X264), X260)))).\nand substitution\nX258 -> T163\nX259 -> T164\nT94 -> .(T163, T164)\nX13 -> X265\nX260 -> X265", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 206 [arrowhead = none ];
206 -> 87;

207 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 207 [arrowhead = none ];
207 -> 88;

208 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 208 [arrowhead = none ];
208 -> 86;

209 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
87 -> 209 [arrowhead = none ];
209 -> 89;

210 [label="SPLIT 2\nnew knowledge:\nT167 is ground\nT168 is ground\nreplacements:\nX261 -> T167\nX262 -> T168", fontsize=14, style = filled, fillcolor = violet];
87 -> 210 [arrowhead = none ];
210 -> 90;

211 [label="INSTANCE with matching:\nT68 -> T164\nT69 -> T163\nX93 -> X261\nX94 -> X262", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 211 [arrowhead = none , style=dashed];
211 -> 51[style=dashed];

212 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
90 -> 212 [arrowhead = none ];
212 -> 91;

213 [label="SPLIT 2\nnew knowledge:\nT171 is ground\nreplacements:\nX263 -> T171", fontsize=14, style = filled, fillcolor = violet];
90 -> 213 [arrowhead = none ];
213 -> 92;

214 [label="INSTANCE with matching:\nT94 -> T167\nX13 -> X263", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 214 [arrowhead = none , style=dashed];
214 -> 79[style=dashed];

215 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
92 -> 215 [arrowhead = none ];
215 -> 93;

216 [label="SPLIT 2\nnew knowledge:\nT172 is ground\nreplacements:\nX264 -> T172", fontsize=14, style = filled, fillcolor = violet];
92 -> 216 [arrowhead = none ];
216 -> 94;

217 [label="INSTANCE with matching:\nT94 -> T168\nX13 -> X264", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 217 [arrowhead = none , style=dashed];
217 -> 79[style=dashed];

218 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 218 [arrowhead = none ];
218 -> 95;

219 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
95 -> 219 [arrowhead = none ];
219 -> 96;

220 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
95 -> 220 [arrowhead = none ];
220 -> 97;

221 [label="EVAL with clause\nappend([], X279, X279).\nand substitution\nT171 -> []\nT163 -> T185\nT172 -> T186\nX279 -> .(T185, T186)\nX265 -> .(T185, T186)", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 221 [arrowhead = none ];
221 -> 98;

222 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 222 [arrowhead = none ];
222 -> 99;

223 [label="EVAL with clause\nappend(.(X291, X292), X293, .(X291, X294)) :- append(X292, X293, X294).\nand substitution\nX291 -> T195\nX292 -> T196\nT171 -> .(T195, T196)\nT163 -> T197\nT172 -> T198\nX293 -> .(T197, T198)\nX294 -> X295\nX265 -> .(T195, X295)", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 223 [arrowhead = none ];
223 -> 101;

224 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 224 [arrowhead = none ];
224 -> 102;

225 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
98 -> 225 [arrowhead = none ];
225 -> 100;

226 [label="INSTANCE with matching:\nT171 -> T196\nT163 -> T197\nT172 -> T198\nX265 -> X295", fontsize=14, style = filled, fillcolor = lightgrey];
101 -> 226 [arrowhead = none , style=dashed];
226 -> 94[style=dashed];

227 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
103 -> 227 [arrowhead = none ];
227 -> 105;

228 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
103 -> 228 [arrowhead = none ];
228 -> 106;

229 [label="INSTANCE with matching:\nT129 -> T213\nT131 -> T215", fontsize=14, style = filled, fillcolor = lightgrey];
105 -> 229 [arrowhead = none , style=dashed];
229 -> 67[style=dashed];

230 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
106 -> 230 [arrowhead = none ];
230 -> 107;

231 [label="SPLIT 2\nnew knowledge:\nT222 is ground\nT223 is ground\nreplacements:\nX323 -> T222\nX324 -> T223", fontsize=14, style = filled, fillcolor = violet];
106 -> 231 [arrowhead = none ];
231 -> 108;

232 [label="INSTANCE with matching:\nT68 -> T214\nT69 -> T215\nX93 -> X323\nX94 -> X324", fontsize=14, style = filled, fillcolor = lightgrey];
107 -> 232 [arrowhead = none , style=dashed];
232 -> 51[style=dashed];

233 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
108 -> 233 [arrowhead = none ];
233 -> 109;

234 [label="SPLIT 2\nnew knowledge:\nT226 is ground\nreplacements:\nX12 -> T226", fontsize=14, style = filled, fillcolor = violet];
108 -> 234 [arrowhead = none ];
234 -> 110;

235 [label="INSTANCE with matching:\nT94 -> T222\nX13 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 235 [arrowhead = none , style=dashed];
235 -> 79[style=dashed];

236 [label="INSTANCE with matching:\nT94 -> .(T213, T223)\nT155 -> T226\nT69 -> T215", fontsize=14, style = filled, fillcolor = lightgrey];
110 -> 236 [arrowhead = none , style=dashed];
236 -> 78[style=dashed];

}

(2) Obligation:

Clauses:

qsort14([]).
append23([], T33, T34, .(T33, T34)).
append23(.(T45, T46), T47, T48, .(T45, T50)) :- append23(T46, T47, T48, T50).
le36(s(T82), s(T83)) :- le36(T82, T83).
le36(0, s(T90)).
le36(0, 0).
split51([], T101, [], []).
split51(.(T114, T115), T116, .(T114, X167), X168) :- le36(T114, T116).
split51(.(T114, T115), T116, .(T114, X167), X168) :- ','(le36(T114, T116), split51(T115, T116, X167, X168)).
split51(.(T129, T130), T131, X199, .(T129, X200)) :- gt67(T129, T131).
split51(.(T129, T130), T131, X199, .(T129, X200)) :- ','(gt67(T129, T131), split51(T130, T131, X199, X200)).
gt67(s(T144), s(T145)) :- gt67(T144, T145).
gt67(s(T150), 0).
qsort79([], []).
qsort79(.(T163, T164), X265) :- split51(T164, T163, X261, X262).
qsort79(.(T163, T164), X265) :- ','(split51(T164, T163, T167, T168), qsort79(T167, X263)).
qsort79(.(T163, T164), X265) :- ','(split51(T164, T163, T167, T168), ','(qsort79(T167, T171), qsort79(T168, X264))).
qsort79(.(T163, T164), X265) :- ','(split51(T164, T163, T167, T168), ','(qsort79(T167, T171), ','(qsort79(T168, T172), append94(T171, T163, T172, X265)))).
append94([], T185, T186, .(T185, T186)).
append94(.(T195, T196), T197, T198, .(T195, X295)) :- append94(T196, T197, T198, X295).
p78(T94, X13, T155, T69, T9) :- qsort79(T94, X13).
p78(T94, T158, T155, T69, T9) :- ','(qsort79(T94, T158), append23(T155, T69, T158, T9)).
qsort1([], []).
qsort1(.(T14, []), T9) :- qsort14(X12).
qsort1(.(T14, []), T9) :- ','(qsort14(T17), qsort14(X13)).
qsort1(.(T14, []), T9) :- ','(qsort14(T17), ','(qsort14(T20), append23(T17, T14, T20, T9))).
qsort1(.(T69, .(T67, T68)), T9) :- le36(T67, T69).
qsort1(.(T69, .(T67, T68)), T9) :- ','(le36(T67, T69), split51(T68, T69, X93, X94)).
qsort1(.(T69, .(T67, T68)), T9) :- ','(le36(T67, T69), ','(split51(T68, T69, T93, T94), qsort1(.(T67, T93), X12))).
qsort1(.(T69, .(T67, T68)), T9) :- ','(le36(T67, T69), ','(split51(T68, T69, T93, T94), ','(qsort1(.(T67, T93), T155), p78(T94, X13, T155, T69, T9)))).
qsort1(.(T215, .(T213, T214)), T9) :- gt67(T213, T215).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gt67(T213, T215), split51(T214, T215, X323, X324)).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gt67(T213, T215), ','(split51(T214, T215, T222, T223), qsort79(T222, X12))).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gt67(T213, T215), ','(split51(T214, T215, T222, T223), ','(qsort79(T222, T226), p78(.(T213, T223), X13, T226, T215, T9)))).

Queries:

qsort1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qsort1_in: (b,f) (f,f)
append23_in: (b,b,b,f) (b,f,b,f) (f,b,f,f) (f,f,f,f)
le36_in: (b,b) (f,f) (f,b)
split51_in: (b,b,f,f) (f,f,f,f) (f,b,f,f)
gt67_in: (b,b) (f,f) (f,b)
qsort79_in: (f,f)
append94_in: (f,f,f,f)
p78_in: (f,f,f,b,f) (f,f,f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)

(5) 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:

QSORT1_IN_GA(.(T14, []), T9) → U19_GA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_GA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_GA(.(T14, []), T9) → U20_GA(T14, T9, qsort14_in_a(T17))
U20_GA(T14, T9, qsort14_out_a(T17)) → U21_GA(T14, T9, qsort14_in_a(X13))
U20_GA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_GA(T14, T9, qsort14_out_a(T17)) → U22_GA(T14, T9, T17, qsort14_in_a(T20))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, le36_in_gg(T67, T69))
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → LE36_IN_GG(T67, T69)
LE36_IN_GG(s(T82), s(T83)) → U2_GG(T82, T83, le36_in_gg(T82, T83))
LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → SPLIT51_IN_GGAA(T68, T69, X93, X94)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_GG(T114, T116)
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_GG(T129, T131)
GT67_IN_GG(s(T144), s(T145)) → U7_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_GA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T14, []), T9) → U19_AA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_AA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_AA(.(T14, []), T9) → U20_AA(T14, T9, qsort14_in_a(T17))
U20_AA(T14, T9, qsort14_out_a(T17)) → U21_AA(T14, T9, qsort14_in_a(X13))
U20_AA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_AA(T14, T9, qsort14_out_a(T17)) → U22_AA(T14, T9, T17, qsort14_in_a(T20))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → LE36_IN_AA(T67, T69)
LE36_IN_AA(s(T82), s(T83)) → U2_AA(T82, T83, le36_in_aa(T82, T83))
LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → SPLIT51_IN_AAAA(T68, T69, X93, X94)
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AA(T114, T116)
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_AAAA(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AAAA(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AA(T129, T131)
GT67_IN_AA(s(T144), s(T145)) → U7_AA(T144, T145, gt67_in_aa(T144, T145))
GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_AAAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AG(T114, T116)
LE36_IN_AG(s(T82), s(T83)) → U2_AG(T82, T83, le36_in_ag(T82, T83))
LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_AGAA(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AG(T129, T131)
GT67_IN_AG(s(T144), s(T145)) → U7_AG(T144, T145, gt67_in_ag(T144, T145))
GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_AGAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_AA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → U30_AA(T215, T213, T214, T9, gt67_in_aa(T213, T215))
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → GT67_IN_AA(T213, T215)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → SPLIT51_IN_AGAA(T214, T215, X323, X324)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_AA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
QSORT79_IN_AA(.(T163, T164), X265) → U8_AA(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
QSORT79_IN_AA(.(T163, T164), X265) → SPLIT51_IN_AAAA(T164, T163, X261, X262)
QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_AA(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_AA(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_AA(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_AA(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → APPEND94_IN_AAAA(T171, T163, T172, X265)
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → U15_AAAA(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_AA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_AA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_AAAGA(T94, X13, T155, T69, T9) → U16_AAAGA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAGA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAGA(T94, T158, T155, T69, T9) → U17_AAAGA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAGA(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AGAA(T155, T69, T158, T9)
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_AA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAAA(T94, X13, T155, T69, T9)
P78_IN_AAAAA(T94, X13, T155, T69, T9) → U16_AAAAA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAAA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAAA(T94, T158, T155, T69, T9) → U17_AAAAA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAAA(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AAAA(T155, T69, T158, T9)
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAGA(T94, X13, T155, T69, T9)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U30_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_GA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_GA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_GA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x3)
QSORT14_IN_A(x1)  =  QSORT14_IN_A
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x3)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x3, x4)
U23_GA(x1, x2, x3)  =  U23_GA(x3)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x3, x5)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x2, x3, x6)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x6)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x5)
QSORT1_IN_AA(x1, x2)  =  QSORT1_IN_AA
U19_AA(x1, x2, x3)  =  U19_AA(x3)
U20_AA(x1, x2, x3)  =  U20_AA(x3)
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x3, x4)
U23_AA(x1, x2, x3)  =  U23_AA(x3)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)
U1_GAGA(x1, x2, x3, x4, x5, x6)  =  U1_GAGA(x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U4_AGAA(x1, x2, x3, x4, x5, x6)  =  U4_AGAA(x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U6_AGAA(x1, x2, x3, x4, x5, x6)  =  U6_AGAA(x6)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
U31_AA(x1, x2, x3, x4, x5)  =  U31_AA(x5)
U32_AA(x1, x2, x3, x4, x5)  =  U32_AA(x1, x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4)  =  U10_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4)  =  U12_AA(x4)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4)  =  U14_AA(x4)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA
U15_AAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAA(x6)
U34_AA(x1, x2, x3, x4, x5, x6)  =  U34_AA(x1, x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(x5)
P78_IN_AAAGA(x1, x2, x3, x4, x5)  =  P78_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x6)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x6)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
P78_IN_AAAAA(x1, x2, x3, x4, x5)  =  P78_IN_AAAAA
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
U17_AAAAA(x1, x2, x3, x4, x5, x6)  =  U17_AAAAA(x6)
U18_AAAAA(x1, x2, x3, x4, x5, x6)  =  U18_AAAAA(x6)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)
U28_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x5)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x3, x5)
U31_GA(x1, x2, x3, x4, x5)  =  U31_GA(x5)
U32_GA(x1, x2, x3, x4, x5)  =  U32_GA(x1, x5)
U33_GA(x1, x2, x3, x4, x5)  =  U33_GA(x5)
U34_GA(x1, x2, x3, x4, x5, x6)  =  U34_GA(x1, x6)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x5)

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

(6) Obligation:

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

QSORT1_IN_GA(.(T14, []), T9) → U19_GA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_GA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_GA(.(T14, []), T9) → U20_GA(T14, T9, qsort14_in_a(T17))
U20_GA(T14, T9, qsort14_out_a(T17)) → U21_GA(T14, T9, qsort14_in_a(X13))
U20_GA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_GA(T14, T9, qsort14_out_a(T17)) → U22_GA(T14, T9, T17, qsort14_in_a(T20))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, le36_in_gg(T67, T69))
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → LE36_IN_GG(T67, T69)
LE36_IN_GG(s(T82), s(T83)) → U2_GG(T82, T83, le36_in_gg(T82, T83))
LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → SPLIT51_IN_GGAA(T68, T69, X93, X94)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_GG(T114, T116)
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_GG(T129, T131)
GT67_IN_GG(s(T144), s(T145)) → U7_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_GA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T14, []), T9) → U19_AA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_AA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_AA(.(T14, []), T9) → U20_AA(T14, T9, qsort14_in_a(T17))
U20_AA(T14, T9, qsort14_out_a(T17)) → U21_AA(T14, T9, qsort14_in_a(X13))
U20_AA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_AA(T14, T9, qsort14_out_a(T17)) → U22_AA(T14, T9, T17, qsort14_in_a(T20))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → LE36_IN_AA(T67, T69)
LE36_IN_AA(s(T82), s(T83)) → U2_AA(T82, T83, le36_in_aa(T82, T83))
LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → SPLIT51_IN_AAAA(T68, T69, X93, X94)
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AA(T114, T116)
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_AAAA(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AAAA(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AA(T129, T131)
GT67_IN_AA(s(T144), s(T145)) → U7_AA(T144, T145, gt67_in_aa(T144, T145))
GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_AAAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AG(T114, T116)
LE36_IN_AG(s(T82), s(T83)) → U2_AG(T82, T83, le36_in_ag(T82, T83))
LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_AGAA(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AG(T129, T131)
GT67_IN_AG(s(T144), s(T145)) → U7_AG(T144, T145, gt67_in_ag(T144, T145))
GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_AGAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_AA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → U30_AA(T215, T213, T214, T9, gt67_in_aa(T213, T215))
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → GT67_IN_AA(T213, T215)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → SPLIT51_IN_AGAA(T214, T215, X323, X324)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_AA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
QSORT79_IN_AA(.(T163, T164), X265) → U8_AA(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
QSORT79_IN_AA(.(T163, T164), X265) → SPLIT51_IN_AAAA(T164, T163, X261, X262)
QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_AA(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_AA(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_AA(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_AA(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → APPEND94_IN_AAAA(T171, T163, T172, X265)
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → U15_AAAA(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_AA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_AA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_AAAGA(T94, X13, T155, T69, T9) → U16_AAAGA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAGA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAGA(T94, T158, T155, T69, T9) → U17_AAAGA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAGA(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AGAA(T155, T69, T158, T9)
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_AA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAAA(T94, X13, T155, T69, T9)
P78_IN_AAAAA(T94, X13, T155, T69, T9) → U16_AAAAA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAAA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAAA(T94, T158, T155, T69, T9) → U17_AAAAA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAAA(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AAAA(T155, T69, T158, T9)
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAGA(T94, X13, T155, T69, T9)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U30_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_GA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_GA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_GA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x3)
QSORT14_IN_A(x1)  =  QSORT14_IN_A
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x3)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x3, x4)
U23_GA(x1, x2, x3)  =  U23_GA(x3)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x3, x5)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x2, x3, x6)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x6)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x5)
QSORT1_IN_AA(x1, x2)  =  QSORT1_IN_AA
U19_AA(x1, x2, x3)  =  U19_AA(x3)
U20_AA(x1, x2, x3)  =  U20_AA(x3)
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x3, x4)
U23_AA(x1, x2, x3)  =  U23_AA(x3)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)
U1_GAGA(x1, x2, x3, x4, x5, x6)  =  U1_GAGA(x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U4_AGAA(x1, x2, x3, x4, x5, x6)  =  U4_AGAA(x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U6_AGAA(x1, x2, x3, x4, x5, x6)  =  U6_AGAA(x6)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
U31_AA(x1, x2, x3, x4, x5)  =  U31_AA(x5)
U32_AA(x1, x2, x3, x4, x5)  =  U32_AA(x1, x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4)  =  U10_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4)  =  U12_AA(x4)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4)  =  U14_AA(x4)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA
U15_AAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAA(x6)
U34_AA(x1, x2, x3, x4, x5, x6)  =  U34_AA(x1, x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(x5)
P78_IN_AAAGA(x1, x2, x3, x4, x5)  =  P78_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x6)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x6)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
P78_IN_AAAAA(x1, x2, x3, x4, x5)  =  P78_IN_AAAAA
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
U17_AAAAA(x1, x2, x3, x4, x5, x6)  =  U17_AAAAA(x6)
U18_AAAAA(x1, x2, x3, x4, x5, x6)  =  U18_AAAAA(x6)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)
U28_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x5)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x3, x5)
U31_GA(x1, x2, x3, x4, x5)  =  U31_GA(x5)
U32_GA(x1, x2, x3, x4, x5)  =  U32_GA(x1, x5)
U33_GA(x1, x2, x3, x4, x5)  =  U33_GA(x5)
U34_GA(x1, x2, x3, x4, x5, x6)  =  U34_GA(x1, x6)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 16 SCCs with 95 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APPEND23_IN_AAAA → APPEND23_IN_AAAA

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APPEND23_IN_AAAA evaluates to t =APPEND23_IN_AAAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND23_IN_AAAA to APPEND23_IN_AAAA.

(15) NO

(16) Obligation:

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

APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APPEND23_IN_AGAA(T47) → APPEND23_IN_AGAA(T47)

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

(21) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APPEND23_IN_AGAA(T47) evaluates to t =APPEND23_IN_AGAA(T47)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND23_IN_AGAA(T47) to APPEND23_IN_AGAA(T47).

(22) NO

(23) Obligation:

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

APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APPEND94_IN_AAAA → APPEND94_IN_AAAA

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

(28) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APPEND94_IN_AAAA evaluates to t =APPEND94_IN_AAAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND94_IN_AAAA to APPEND94_IN_AAAA.

(29) NO

(30) Obligation:

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

GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

GT67_IN_AG(s(T145)) → GT67_IN_AG(T145)

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

(35) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• GT67_IN_AG(s(T145)) → GT67_IN_AG(T145)
  The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

LE36_IN_AG(s(T83)) → LE36_IN_AG(T83)

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

(42) 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:

• LE36_IN_AG(s(T83)) → LE36_IN_AG(T83)
  The graph contains the following edges 1 > 1

(43) YES

(44) Obligation:

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

U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114)) → SPLIT51_IN_AGAA(T116)
SPLIT51_IN_AGAA(T116) → U3_AGAA(T116, le36_in_ag(T116))
SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131))
U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(49) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AGAA(T116) → U3_AGAA(T116, le36_in_ag(T116)) at position [1] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))

(50) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114)) → SPLIT51_IN_AGAA(T116)
SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131))
U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(51) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131)) at position [1] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)

(52) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114)) → SPLIT51_IN_AGAA(T116)
U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(53) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGAA(T116, le36_out_ag(T114)) → SPLIT51_IN_AGAA(T116) we obtained the following new rules [LPAR04]:

U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)

(54) Obligation:

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

U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(55) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131) we obtained the following new rules [LPAR04]:

U5_AGAA(s(z0), gt67_out_ag) → SPLIT51_IN_AGAA(s(z0))
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

(56) Obligation:

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

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)
U5_AGAA(s(z0), gt67_out_ag) → SPLIT51_IN_AGAA(s(z0))
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(57) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(58) Complex Obligation (AND)

(59) Obligation:

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

U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(60) UsableRulesProof (EQUIVALENT transformation)

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

(61) Obligation:

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

U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

R is empty.
The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(62) QReductionProof (EQUIVALENT transformation)

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

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

(63) Obligation:

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

U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

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

(64) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = SPLIT51_IN_AGAA(0) evaluates to t =SPLIT51_IN_AGAA(0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
with rule SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0)) at position [] and matcher [ ]

U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)
with rule U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(65) NO

(66) Obligation:

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

U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
U3_AGAA(s(z0), le36_out_ag(0)) → SPLIT51_IN_AGAA(s(z0))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
U5_AGAA(s(z0), gt67_out_ag) → SPLIT51_IN_AGAA(s(z0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(67) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = SPLIT51_IN_AGAA(s(x0)) evaluates to t =SPLIT51_IN_AGAA(s(x0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
with rule SPLIT51_IN_AGAA(s(x0')) → U3_AGAA(s(x0'), le36_out_ag(0)) at position [] and matcher [x0' / x0]

U3_AGAA(s(x0), le36_out_ag(0)) → SPLIT51_IN_AGAA(s(x0))
with rule U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(68) NO

(69) Obligation:

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

GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA

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

(70) UsableRulesProof (EQUIVALENT transformation)

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

(71) Obligation:

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

GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA

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

(72) PiDPToQDPProof (SOUND transformation)

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

(73) Obligation:

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

GT67_IN_AA → GT67_IN_AA

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

(74) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = GT67_IN_AA evaluates to t =GT67_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT67_IN_AA to GT67_IN_AA.

(75) NO

(76) Obligation:

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

LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA

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

(77) UsableRulesProof (EQUIVALENT transformation)

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

(78) Obligation:

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

LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA

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

(79) PiDPToQDPProof (SOUND transformation)

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

(80) Obligation:

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

LE36_IN_AA → LE36_IN_AA

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

(81) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = LE36_IN_AA evaluates to t =LE36_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE36_IN_AA to LE36_IN_AA.

(82) NO

(83) Obligation:

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

SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)

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

(84) UsableRulesProof (EQUIVALENT transformation)

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

(85) Obligation:

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

SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)

The TRS R consists of the following rules:

le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)

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

(86) PiDPToQDPProof (SOUND transformation)

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

(87) Obligation:

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

SPLIT51_IN_AAAA → U3_AAAA(le36_in_aa)
U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))

The set Q consists of the following terms:

le36_in_aa
U2_aa(x0)

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

(88) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AAAA → U3_AAAA(le36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AAAA → U3_AAAA(U2_aa(le36_in_aa))
SPLIT51_IN_AAAA → U3_AAAA(le36_out_aa(0))

(89) Obligation:

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

U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA
SPLIT51_IN_AAAA → U3_AAAA(U2_aa(le36_in_aa))
SPLIT51_IN_AAAA → U3_AAAA(le36_out_aa(0))

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))

The set Q consists of the following terms:

le36_in_aa
U2_aa(x0)

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

(90) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = SPLIT51_IN_AAAA evaluates to t =SPLIT51_IN_AAAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

SPLIT51_IN_AAAA → U3_AAAA(le36_out_aa(0))
with rule SPLIT51_IN_AAAA → U3_AAAA(le36_out_aa(0)) at position [] and matcher [ ]

U3_AAAA(le36_out_aa(0)) → SPLIT51_IN_AAAA
with rule U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(91) NO

(92) Obligation:

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

QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_IN_AA
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)

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

(93) UsableRulesProof (EQUIVALENT transformation)

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

(94) Obligation:

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

QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)

The TRS R consists of the following rules:

split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
QSORT79_IN_AA(x1, x2)  =  QSORT79_IN_AA
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)

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

(95) PiDPToQDPProof (SOUND transformation)

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

(96) Obligation:

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

QSORT79_IN_AA → U9_AA(split51_in_aaaa)
U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa)
U11_AA(qsort79_out_aa) → QSORT79_IN_AA

The TRS R consists of the following rules:

split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
qsort79_in_aa → qsort79_out_aa
qsort79_in_aa → U8_aa(split51_in_aaaa)
qsort79_in_aa → U9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
append94_in_aaaa → append94_out_aaaa
append94_in_aaaa → U15_aaaa(append94_in_aaaa)
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
append94_in_aaaa
U2_ag(x0)
U7_ag(x0)
U15_aaaa(x0)

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

(97) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QSORT79_IN_AA → U9_AA(split51_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QSORT79_IN_AA → U9_AA(split51_out_aaaa)
QSORT79_IN_AA → U9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AA → U9_AA(U5_aaaa(gt67_in_aa))

(98) Obligation:

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

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa)
U11_AA(qsort79_out_aa) → QSORT79_IN_AA
QSORT79_IN_AA → U9_AA(split51_out_aaaa)
QSORT79_IN_AA → U9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AA → U9_AA(U5_aaaa(gt67_in_aa))

The TRS R consists of the following rules:

split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
qsort79_in_aa → qsort79_out_aa
qsort79_in_aa → U8_aa(split51_in_aaaa)
qsort79_in_aa → U9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
append94_in_aaaa → append94_out_aaaa
append94_in_aaaa → U15_aaaa(append94_in_aaaa)
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
append94_in_aaaa
U2_ag(x0)
U7_ag(x0)
U15_aaaa(x0)

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

(99) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U9_AA(split51_out_aaaa) → U11_AA(qsort79_out_aa)
U9_AA(split51_out_aaaa) → U11_AA(U8_aa(split51_in_aaaa))
U9_AA(split51_out_aaaa) → U11_AA(U9_aa(split51_in_aaaa))

(100) Obligation:

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

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U11_AA(qsort79_out_aa) → QSORT79_IN_AA
QSORT79_IN_AA → U9_AA(split51_out_aaaa)
QSORT79_IN_AA → U9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AA → U9_AA(U5_aaaa(gt67_in_aa))
U9_AA(split51_out_aaaa) → U11_AA(qsort79_out_aa)
U9_AA(split51_out_aaaa) → U11_AA(U8_aa(split51_in_aaaa))
U9_AA(split51_out_aaaa) → U11_AA(U9_aa(split51_in_aaaa))

The TRS R consists of the following rules:

split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
qsort79_in_aa → qsort79_out_aa
qsort79_in_aa → U8_aa(split51_in_aaaa)
qsort79_in_aa → U9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
append94_in_aaaa → append94_out_aaaa
append94_in_aaaa → U15_aaaa(append94_in_aaaa)
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
append94_in_aaaa
U2_ag(x0)
U7_ag(x0)
U15_aaaa(x0)

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

(101) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QSORT79_IN_AA evaluates to t =QSORT79_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QSORT79_IN_AA → U9_AA(split51_out_aaaa)
with rule QSORT79_IN_AA → U9_AA(split51_out_aaaa) at position [] and matcher [ ]

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
with rule U9_AA(split51_out_aaaa) → QSORT79_IN_AA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(102) NO

(103) Obligation:

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

APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)

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

(104) UsableRulesProof (EQUIVALENT transformation)

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

(105) Obligation:

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

APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)

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

(106) PiDPToQDPProof (SOUND transformation)

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

(107) Obligation:

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

APPEND23_IN_GAGA(.(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)

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

(108) 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:

• APPEND23_IN_GAGA(.(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)
  The graph contains the following edges 1 > 1, 2 >= 2

(109) YES

(110) Obligation:

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

QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
QSORT1_IN_AA(x1, x2)  =  QSORT1_IN_AA
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)

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

(111) UsableRulesProof (EQUIVALENT transformation)

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

(112) Obligation:

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

QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)

The TRS R consists of the following rules:

le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
QSORT1_IN_AA(x1, x2)  =  QSORT1_IN_AA
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)

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

(113) PiDPToQDPProof (SOUND transformation)

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

(114) Obligation:

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

QSORT1_IN_AA → U24_AA(le36_in_aa)
U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa)
U26_AA(split51_out_aaaa) → QSORT1_IN_AA

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
U2_ag(x0)
U7_ag(x0)

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

(115) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QSORT1_IN_AA → U24_AA(le36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QSORT1_IN_AA → U24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AA → U24_AA(le36_out_aa(0))

(116) Obligation:

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

U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa)
U26_AA(split51_out_aaaa) → QSORT1_IN_AA
QSORT1_IN_AA → U24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AA → U24_AA(le36_out_aa(0))

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
U2_ag(x0)
U7_ag(x0)

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

(117) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa)
U24_AA(le36_out_aa(y0)) → U26_AA(U3_aaaa(le36_in_aa))
U24_AA(le36_out_aa(y0)) → U26_AA(U5_aaaa(gt67_in_aa))

(118) Obligation:

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

U26_AA(split51_out_aaaa) → QSORT1_IN_AA
QSORT1_IN_AA → U24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AA → U24_AA(le36_out_aa(0))
U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa)
U24_AA(le36_out_aa(y0)) → U26_AA(U3_aaaa(le36_in_aa))
U24_AA(le36_out_aa(y0)) → U26_AA(U5_aaaa(gt67_in_aa))

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
U2_ag(x0)
U7_ag(x0)

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

(119) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QSORT1_IN_AA evaluates to t =QSORT1_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

QSORT1_IN_AA → U24_AA(le36_out_aa(0))
with rule QSORT1_IN_AA → U24_AA(le36_out_aa(0)) at position [] and matcher [ ]

U24_AA(le36_out_aa(0)) → U26_AA(split51_out_aaaa)
with rule U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa) at position [] and matcher [y0 / 0]

U26_AA(split51_out_aaaa) → QSORT1_IN_AA
with rule U26_AA(split51_out_aaaa) → QSORT1_IN_AA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(120) NO

(121) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)

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

(122) UsableRulesProof (EQUIVALENT transformation)

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

(123) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

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

(124) PiDPToQDPProof (EQUIVALENT transformation)

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

(125) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

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

(126) 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:

• GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
  The graph contains the following edges 1 > 1, 2 > 2

(127) YES

(128) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)

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

(129) UsableRulesProof (EQUIVALENT transformation)

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

(130) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

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

(131) PiDPToQDPProof (EQUIVALENT transformation)

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

(132) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

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

(133) 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:

• LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
  The graph contains the following edges 1 > 1, 2 > 2

(134) YES

(135) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x2, x3, x6)

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

(136) UsableRulesProof (EQUIVALENT transformation)

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

(137) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x2, x3, x6)

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

(138) PiDPToQDPProof (SOUND transformation)

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

(139) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116) → U3_GGAA(T115, T116, le36_in_gg(T114, T116))
U3_GGAA(T115, T116, le36_out_gg) → SPLIT51_IN_GGAA(T115, T116)
SPLIT51_IN_GGAA(.(T129, T130), T131) → U5_GGAA(T130, T131, gt67_in_gg(T129, T131))
U5_GGAA(T130, T131, gt67_out_gg) → SPLIT51_IN_GGAA(T130, T131)

The TRS R consists of the following rules:

le36_in_gg(s(T82), s(T83)) → U2_gg(le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg
le36_in_gg(0, 0) → le36_out_gg
gt67_in_gg(s(T144), s(T145)) → U7_gg(gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg
U2_gg(le36_out_gg) → le36_out_gg
U7_gg(gt67_out_gg) → gt67_out_gg

The set Q consists of the following terms:

le36_in_gg(x0, x1)
gt67_in_gg(x0, x1)
U2_gg(x0)
U7_gg(x0)

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

(140) 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:

• U3_GGAA(T115, T116, le36_out_gg) → SPLIT51_IN_GGAA(T115, T116)
  The graph contains the following edges 1 >= 1, 2 >= 2

• U5_GGAA(T130, T131, gt67_out_gg) → SPLIT51_IN_GGAA(T130, T131)
  The graph contains the following edges 1 >= 1, 2 >= 2

• SPLIT51_IN_GGAA(.(T114, T115), T116) → U3_GGAA(T115, T116, le36_in_gg(T114, T116))
  The graph contains the following edges 1 > 1, 2 >= 2

• SPLIT51_IN_GGAA(.(T129, T130), T131) → U5_GGAA(T130, T131, gt67_in_gg(T129, T131))
  The graph contains the following edges 1 > 1, 2 >= 2

(141) YES

(142) Obligation:

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

APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)

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

(143) UsableRulesProof (EQUIVALENT transformation)

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

(144) Obligation:

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

APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)

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

(145) PiDPToQDPProof (SOUND transformation)

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

(146) Obligation:

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

APPEND23_IN_GGGA(.(T45, T46), T47, T48) → APPEND23_IN_GGGA(T46, T47, T48)

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

(147) 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:

• APPEND23_IN_GGGA(.(T45, T46), T47, T48) → APPEND23_IN_GGGA(T46, T47, T48)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(148) YES

(149) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qsort1_in: (b,f) (f,f)
append23_in: (b,b,b,f) (b,f,b,f) (f,b,f,f) (f,f,f,f)
le36_in: (b,b) (f,f) (f,b)
split51_in: (b,b,f,f) (f,f,f,f) (f,b,f,f)
gt67_in: (b,b) (f,f) (f,b)
qsort79_in: (f,f)
append94_in: (f,f,f,f)
p78_in: (f,f,f,b,f) (f,f,f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(150) Obligation:

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

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)

(151) 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:

QSORT1_IN_GA(.(T14, []), T9) → U19_GA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_GA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_GA(.(T14, []), T9) → U20_GA(T14, T9, qsort14_in_a(T17))
U20_GA(T14, T9, qsort14_out_a(T17)) → U21_GA(T14, T9, qsort14_in_a(X13))
U20_GA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_GA(T14, T9, qsort14_out_a(T17)) → U22_GA(T14, T9, T17, qsort14_in_a(T20))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, le36_in_gg(T67, T69))
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → LE36_IN_GG(T67, T69)
LE36_IN_GG(s(T82), s(T83)) → U2_GG(T82, T83, le36_in_gg(T82, T83))
LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → SPLIT51_IN_GGAA(T68, T69, X93, X94)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_GG(T114, T116)
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_GG(T129, T131)
GT67_IN_GG(s(T144), s(T145)) → U7_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_GA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T14, []), T9) → U19_AA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_AA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_AA(.(T14, []), T9) → U20_AA(T14, T9, qsort14_in_a(T17))
U20_AA(T14, T9, qsort14_out_a(T17)) → U21_AA(T14, T9, qsort14_in_a(X13))
U20_AA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_AA(T14, T9, qsort14_out_a(T17)) → U22_AA(T14, T9, T17, qsort14_in_a(T20))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → LE36_IN_AA(T67, T69)
LE36_IN_AA(s(T82), s(T83)) → U2_AA(T82, T83, le36_in_aa(T82, T83))
LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → SPLIT51_IN_AAAA(T68, T69, X93, X94)
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AA(T114, T116)
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_AAAA(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AAAA(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AA(T129, T131)
GT67_IN_AA(s(T144), s(T145)) → U7_AA(T144, T145, gt67_in_aa(T144, T145))
GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_AAAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AG(T114, T116)
LE36_IN_AG(s(T82), s(T83)) → U2_AG(T82, T83, le36_in_ag(T82, T83))
LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_AGAA(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AG(T129, T131)
GT67_IN_AG(s(T144), s(T145)) → U7_AG(T144, T145, gt67_in_ag(T144, T145))
GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_AGAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_AA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → U30_AA(T215, T213, T214, T9, gt67_in_aa(T213, T215))
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → GT67_IN_AA(T213, T215)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → SPLIT51_IN_AGAA(T214, T215, X323, X324)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_AA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
QSORT79_IN_AA(.(T163, T164), X265) → U8_AA(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
QSORT79_IN_AA(.(T163, T164), X265) → SPLIT51_IN_AAAA(T164, T163, X261, X262)
QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_AA(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_AA(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_AA(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_AA(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → APPEND94_IN_AAAA(T171, T163, T172, X265)
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → U15_AAAA(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_AA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_AA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_AAAGA(T94, X13, T155, T69, T9) → U16_AAAGA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAGA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAGA(T94, T158, T155, T69, T9) → U17_AAAGA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAGA(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AGAA(T155, T69, T158, T9)
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_AA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAAA(T94, X13, T155, T69, T9)
P78_IN_AAAAA(T94, X13, T155, T69, T9) → U16_AAAAA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAAA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAAA(T94, T158, T155, T69, T9) → U17_AAAAA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAAA(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AAAA(T155, T69, T158, T9)
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAGA(T94, X13, T155, T69, T9)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U30_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_GA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_GA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_GA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
QSORT14_IN_A(x1)  =  QSORT14_IN_A
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x3, x4)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
QSORT1_IN_AA(x1, x2)  =  QSORT1_IN_AA
U19_AA(x1, x2, x3)  =  U19_AA(x3)
U20_AA(x1, x2, x3)  =  U20_AA(x3)
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x3, x4)
U23_AA(x1, x2, x3)  =  U23_AA(x3)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)
U1_GAGA(x1, x2, x3, x4, x5, x6)  =  U1_GAGA(x1, x2, x4, x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U4_AGAA(x1, x2, x3, x4, x5, x6)  =  U4_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
U6_AGAA(x1, x2, x3, x4, x5, x6)  =  U6_AGAA(x3, x6)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
U31_AA(x1, x2, x3, x4, x5)  =  U31_AA(x5)
U32_AA(x1, x2, x3, x4, x5)  =  U32_AA(x1, x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4)  =  U10_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4)  =  U12_AA(x4)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4)  =  U14_AA(x4)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA
U15_AAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAA(x6)
U34_AA(x1, x2, x3, x4, x5, x6)  =  U34_AA(x1, x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(x5)
P78_IN_AAAGA(x1, x2, x3, x4, x5)  =  P78_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x4, x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x4, x6)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x3, x6)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
P78_IN_AAAAA(x1, x2, x3, x4, x5)  =  P78_IN_AAAAA
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
U17_AAAAA(x1, x2, x3, x4, x5, x6)  =  U17_AAAAA(x6)
U18_AAAAA(x1, x2, x3, x4, x5, x6)  =  U18_AAAAA(x6)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)
U28_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x2, x3, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x1, x2, x3, x5)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x2, x3, x5)
U31_GA(x1, x2, x3, x4, x5)  =  U31_GA(x1, x2, x3, x5)
U32_GA(x1, x2, x3, x4, x5)  =  U32_GA(x1, x2, x3, x5)
U33_GA(x1, x2, x3, x4, x5)  =  U33_GA(x1, x2, x3, x5)
U34_GA(x1, x2, x3, x4, x5, x6)  =  U34_GA(x1, x2, x3, x6)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x1, x2, x3, x5)

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

(152) Obligation:

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

QSORT1_IN_GA(.(T14, []), T9) → U19_GA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_GA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_GA(.(T14, []), T9) → U20_GA(T14, T9, qsort14_in_a(T17))
U20_GA(T14, T9, qsort14_out_a(T17)) → U21_GA(T14, T9, qsort14_in_a(X13))
U20_GA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_GA(T14, T9, qsort14_out_a(T17)) → U22_GA(T14, T9, T17, qsort14_in_a(T20))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, le36_in_gg(T67, T69))
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → LE36_IN_GG(T67, T69)
LE36_IN_GG(s(T82), s(T83)) → U2_GG(T82, T83, le36_in_gg(T82, T83))
LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → SPLIT51_IN_GGAA(T68, T69, X93, X94)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_GG(T114, T116)
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_GG(T129, T131)
GT67_IN_GG(s(T144), s(T145)) → U7_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_GA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T14, []), T9) → U19_AA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_AA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_AA(.(T14, []), T9) → U20_AA(T14, T9, qsort14_in_a(T17))
U20_AA(T14, T9, qsort14_out_a(T17)) → U21_AA(T14, T9, qsort14_in_a(X13))
U20_AA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_AA(T14, T9, qsort14_out_a(T17)) → U22_AA(T14, T9, T17, qsort14_in_a(T20))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → LE36_IN_AA(T67, T69)
LE36_IN_AA(s(T82), s(T83)) → U2_AA(T82, T83, le36_in_aa(T82, T83))
LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → SPLIT51_IN_AAAA(T68, T69, X93, X94)
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AA(T114, T116)
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_AAAA(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AAAA(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AA(T129, T131)
GT67_IN_AA(s(T144), s(T145)) → U7_AA(T144, T145, gt67_in_aa(T144, T145))
GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_AAAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AG(T114, T116)
LE36_IN_AG(s(T82), s(T83)) → U2_AG(T82, T83, le36_in_ag(T82, T83))
LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_AGAA(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AG(T129, T131)
GT67_IN_AG(s(T144), s(T145)) → U7_AG(T144, T145, gt67_in_ag(T144, T145))
GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_AGAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_AA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → U30_AA(T215, T213, T214, T9, gt67_in_aa(T213, T215))
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → GT67_IN_AA(T213, T215)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → SPLIT51_IN_AGAA(T214, T215, X323, X324)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_AA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
QSORT79_IN_AA(.(T163, T164), X265) → U8_AA(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
QSORT79_IN_AA(.(T163, T164), X265) → SPLIT51_IN_AAAA(T164, T163, X261, X262)
QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_AA(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_AA(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_AA(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_AA(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → APPEND94_IN_AAAA(T171, T163, T172, X265)
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → U15_AAAA(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_AA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_AA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_AAAGA(T94, X13, T155, T69, T9) → U16_AAAGA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAGA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAGA(T94, T158, T155, T69, T9) → U17_AAAGA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAGA(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AGAA(T155, T69, T158, T9)
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_AA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAAA(T94, X13, T155, T69, T9)
P78_IN_AAAAA(T94, X13, T155, T69, T9) → U16_AAAAA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAAA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAAA(T94, T158, T155, T69, T9) → U17_AAAAA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAAA(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AAAA(T155, T69, T158, T9)
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAGA(T94, X13, T155, T69, T9)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U30_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_GA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_GA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_GA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
QSORT14_IN_A(x1)  =  QSORT14_IN_A
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x3, x4)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
QSORT1_IN_AA(x1, x2)  =  QSORT1_IN_AA
U19_AA(x1, x2, x3)  =  U19_AA(x3)
U20_AA(x1, x2, x3)  =  U20_AA(x3)
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x3, x4)
U23_AA(x1, x2, x3)  =  U23_AA(x3)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)
U1_GAGA(x1, x2, x3, x4, x5, x6)  =  U1_GAGA(x1, x2, x4, x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U4_AGAA(x1, x2, x3, x4, x5, x6)  =  U4_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
U6_AGAA(x1, x2, x3, x4, x5, x6)  =  U6_AGAA(x3, x6)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
U31_AA(x1, x2, x3, x4, x5)  =  U31_AA(x5)
U32_AA(x1, x2, x3, x4, x5)  =  U32_AA(x1, x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4)  =  U10_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4)  =  U12_AA(x4)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4)  =  U14_AA(x4)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA
U15_AAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAA(x6)
U34_AA(x1, x2, x3, x4, x5, x6)  =  U34_AA(x1, x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(x5)
P78_IN_AAAGA(x1, x2, x3, x4, x5)  =  P78_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x4, x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x4, x6)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x3, x6)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
P78_IN_AAAAA(x1, x2, x3, x4, x5)  =  P78_IN_AAAAA
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
U17_AAAAA(x1, x2, x3, x4, x5, x6)  =  U17_AAAAA(x6)
U18_AAAAA(x1, x2, x3, x4, x5, x6)  =  U18_AAAAA(x6)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)
U28_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x2, x3, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x1, x2, x3, x5)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x2, x3, x5)
U31_GA(x1, x2, x3, x4, x5)  =  U31_GA(x1, x2, x3, x5)
U32_GA(x1, x2, x3, x4, x5)  =  U32_GA(x1, x2, x3, x5)
U33_GA(x1, x2, x3, x4, x5)  =  U33_GA(x1, x2, x3, x5)
U34_GA(x1, x2, x3, x4, x5, x6)  =  U34_GA(x1, x2, x3, x6)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x1, x2, x3, x5)

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

(153) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 16 SCCs with 95 less nodes.

(154) Complex Obligation (AND)

(155) Obligation:

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

APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA

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

(156) UsableRulesProof (EQUIVALENT transformation)

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

(157) Obligation:

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

APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA

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

(158) PiDPToQDPProof (SOUND transformation)

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

(159) Obligation:

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

APPEND23_IN_AAAA → APPEND23_IN_AAAA

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

(160) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APPEND23_IN_AAAA evaluates to t =APPEND23_IN_AAAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND23_IN_AAAA to APPEND23_IN_AAAA.

(161) NO

(162) Obligation:

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

APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)

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

(163) UsableRulesProof (EQUIVALENT transformation)

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

(164) Obligation:

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

APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)

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

(165) PiDPToQDPProof (SOUND transformation)

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

(166) Obligation:

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

APPEND23_IN_AGAA(T47) → APPEND23_IN_AGAA(T47)

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

(167) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APPEND23_IN_AGAA(T47) evaluates to t =APPEND23_IN_AGAA(T47)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND23_IN_AGAA(T47) to APPEND23_IN_AGAA(T47).

(168) NO

(169) Obligation:

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

APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA

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

(170) UsableRulesProof (EQUIVALENT transformation)

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

(171) Obligation:

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

APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA

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

(172) PiDPToQDPProof (SOUND transformation)

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

(173) Obligation:

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

APPEND94_IN_AAAA → APPEND94_IN_AAAA

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

(174) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APPEND94_IN_AAAA evaluates to t =APPEND94_IN_AAAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND94_IN_AAAA to APPEND94_IN_AAAA.

(175) NO

(176) Obligation:

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

GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)

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

(177) UsableRulesProof (EQUIVALENT transformation)

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

(178) Obligation:

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

GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)

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

(179) PiDPToQDPProof (SOUND transformation)

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

(180) Obligation:

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

GT67_IN_AG(s(T145)) → GT67_IN_AG(T145)

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

(181) 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:

• GT67_IN_AG(s(T145)) → GT67_IN_AG(T145)
  The graph contains the following edges 1 > 1

(182) YES

(183) Obligation:

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

LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)

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

(184) UsableRulesProof (EQUIVALENT transformation)

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

(185) Obligation:

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

LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)

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

(186) PiDPToQDPProof (SOUND transformation)

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

(187) Obligation:

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

LE36_IN_AG(s(T83)) → LE36_IN_AG(T83)

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

(188) 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:

• LE36_IN_AG(s(T83)) → LE36_IN_AG(T83)
  The graph contains the following edges 1 > 1

(189) YES

(190) Obligation:

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

U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)

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

(191) UsableRulesProof (EQUIVALENT transformation)

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

(192) Obligation:

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

U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)

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

(193) PiDPToQDPProof (SOUND transformation)

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

(194) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T116)
SPLIT51_IN_AGAA(T116) → U3_AGAA(T116, le36_in_ag(T116))
SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131))
U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(195) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AGAA(T116) → U3_AGAA(T116, le36_in_ag(T116)) at position [1] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))

(196) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T116)
SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131))
U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(197) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131)) at position [1] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))

(198) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T116)
U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(199) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGAA(T116, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T116) we obtained the following new rules [LPAR04]:

U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)

(200) Obligation:

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

U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(201) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131) we obtained the following new rules [LPAR04]:

U5_AGAA(s(z0), gt67_out_ag(s(z0))) → SPLIT51_IN_AGAA(s(z0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

(202) Obligation:

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

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)
U5_AGAA(s(z0), gt67_out_ag(s(z0))) → SPLIT51_IN_AGAA(s(z0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(203) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(204) Complex Obligation (AND)

(205) Obligation:

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

U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(206) UsableRulesProof (EQUIVALENT transformation)

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

(207) Obligation:

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

U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

R is empty.
The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(208) QReductionProof (EQUIVALENT transformation)

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

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

(209) Obligation:

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

U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

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

(210) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = SPLIT51_IN_AGAA(0) evaluates to t =SPLIT51_IN_AGAA(0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
with rule SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0)) at position [] and matcher [ ]

U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)
with rule U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(211) NO

(212) Obligation:

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

U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
U3_AGAA(s(z0), le36_out_ag(0, s(z0))) → SPLIT51_IN_AGAA(s(z0))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
U5_AGAA(s(z0), gt67_out_ag(s(z0))) → SPLIT51_IN_AGAA(s(z0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(213) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = SPLIT51_IN_AGAA(s(x0)) evaluates to t =SPLIT51_IN_AGAA(s(x0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
with rule SPLIT51_IN_AGAA(s(x0')) → U3_AGAA(s(x0'), le36_out_ag(0, s(x0'))) at position [] and matcher [x0' / x0]

U3_AGAA(s(x0), le36_out_ag(0, s(x0))) → SPLIT51_IN_AGAA(s(x0))
with rule U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(214) NO

(215) Obligation:

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

GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA

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

(216) UsableRulesProof (EQUIVALENT transformation)

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

(217) Obligation:

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

GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA

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

(218) PiDPToQDPProof (SOUND transformation)

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

(219) Obligation:

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

GT67_IN_AA → GT67_IN_AA

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

(220) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = GT67_IN_AA evaluates to t =GT67_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT67_IN_AA to GT67_IN_AA.

(221) NO

(222) Obligation:

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

LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA

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

(223) UsableRulesProof (EQUIVALENT transformation)

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

(224) Obligation:

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

LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA

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

(225) PiDPToQDPProof (SOUND transformation)

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

(226) Obligation:

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

LE36_IN_AA → LE36_IN_AA

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

(227) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = LE36_IN_AA evaluates to t =LE36_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE36_IN_AA to LE36_IN_AA.

(228) NO

(229) Obligation:

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

SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)

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

(230) UsableRulesProof (EQUIVALENT transformation)

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

(231) Obligation:

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

SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)

The TRS R consists of the following rules:

le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)

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

(232) PiDPToQDPProof (SOUND transformation)

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

(233) Obligation:

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

SPLIT51_IN_AAAA → U3_AAAA(le36_in_aa)
U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))

The set Q consists of the following terms:

le36_in_aa
U2_aa(x0)

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

(234) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AAAA → U3_AAAA(le36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AAAA → U3_AAAA(U2_aa(le36_in_aa))
SPLIT51_IN_AAAA → U3_AAAA(le36_out_aa(0))

(235) Obligation:

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

U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA
SPLIT51_IN_AAAA → U3_AAAA(U2_aa(le36_in_aa))
SPLIT51_IN_AAAA → U3_AAAA(le36_out_aa(0))

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))

The set Q consists of the following terms:

le36_in_aa
U2_aa(x0)

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

(236) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = SPLIT51_IN_AAAA evaluates to t =SPLIT51_IN_AAAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

SPLIT51_IN_AAAA → U3_AAAA(le36_out_aa(0))
with rule SPLIT51_IN_AAAA → U3_AAAA(le36_out_aa(0)) at position [] and matcher [ ]

U3_AAAA(le36_out_aa(0)) → SPLIT51_IN_AAAA
with rule U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(237) NO

(238) Obligation:

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

QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_IN_AA
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)

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

(239) UsableRulesProof (EQUIVALENT transformation)

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

(240) Obligation:

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

QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)

The TRS R consists of the following rules:

split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
QSORT79_IN_AA(x1, x2)  =  QSORT79_IN_AA
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)

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

(241) PiDPToQDPProof (SOUND transformation)

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

(242) Obligation:

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

QSORT79_IN_AA → U9_AA(split51_in_aaaa)
U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa)
U11_AA(qsort79_out_aa) → QSORT79_IN_AA

The TRS R consists of the following rules:

split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
qsort79_in_aa → qsort79_out_aa
qsort79_in_aa → U8_aa(split51_in_aaaa)
qsort79_in_aa → U9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
append94_in_aaaa → append94_out_aaaa
append94_in_aaaa → U15_aaaa(append94_in_aaaa)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
append94_in_aaaa
U2_ag(x0, x1)
U7_ag(x0, x1)
U15_aaaa(x0)

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

(243) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QSORT79_IN_AA → U9_AA(split51_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QSORT79_IN_AA → U9_AA(split51_out_aaaa)
QSORT79_IN_AA → U9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AA → U9_AA(U5_aaaa(gt67_in_aa))

(244) Obligation:

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

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa)
U11_AA(qsort79_out_aa) → QSORT79_IN_AA
QSORT79_IN_AA → U9_AA(split51_out_aaaa)
QSORT79_IN_AA → U9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AA → U9_AA(U5_aaaa(gt67_in_aa))

The TRS R consists of the following rules:

split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
qsort79_in_aa → qsort79_out_aa
qsort79_in_aa → U8_aa(split51_in_aaaa)
qsort79_in_aa → U9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
append94_in_aaaa → append94_out_aaaa
append94_in_aaaa → U15_aaaa(append94_in_aaaa)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
append94_in_aaaa
U2_ag(x0, x1)
U7_ag(x0, x1)
U15_aaaa(x0)

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

(245) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U9_AA(split51_out_aaaa) → U11_AA(qsort79_out_aa)
U9_AA(split51_out_aaaa) → U11_AA(U8_aa(split51_in_aaaa))
U9_AA(split51_out_aaaa) → U11_AA(U9_aa(split51_in_aaaa))

(246) Obligation:

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

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U11_AA(qsort79_out_aa) → QSORT79_IN_AA
QSORT79_IN_AA → U9_AA(split51_out_aaaa)
QSORT79_IN_AA → U9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AA → U9_AA(U5_aaaa(gt67_in_aa))
U9_AA(split51_out_aaaa) → U11_AA(qsort79_out_aa)
U9_AA(split51_out_aaaa) → U11_AA(U8_aa(split51_in_aaaa))
U9_AA(split51_out_aaaa) → U11_AA(U9_aa(split51_in_aaaa))

The TRS R consists of the following rules:

split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
qsort79_in_aa → qsort79_out_aa
qsort79_in_aa → U8_aa(split51_in_aaaa)
qsort79_in_aa → U9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
append94_in_aaaa → append94_out_aaaa
append94_in_aaaa → U15_aaaa(append94_in_aaaa)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
append94_in_aaaa
U2_ag(x0, x1)
U7_ag(x0, x1)
U15_aaaa(x0)

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

(247) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QSORT79_IN_AA evaluates to t =QSORT79_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

QSORT79_IN_AA → U9_AA(split51_out_aaaa)
with rule QSORT79_IN_AA → U9_AA(split51_out_aaaa) at position [] and matcher [ ]

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
with rule U9_AA(split51_out_aaaa) → QSORT79_IN_AA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(248) NO

(249) Obligation:

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

APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)

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

(250) UsableRulesProof (EQUIVALENT transformation)

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

(251) Obligation:

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

APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)

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

(252) PiDPToQDPProof (SOUND transformation)

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

(253) Obligation:

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

APPEND23_IN_GAGA(.(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)

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

(254) 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:

• APPEND23_IN_GAGA(.(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)
  The graph contains the following edges 1 > 1, 2 >= 2

(255) YES

(256) Obligation:

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

QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
QSORT1_IN_AA(x1, x2)  =  QSORT1_IN_AA
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)

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

(257) UsableRulesProof (EQUIVALENT transformation)

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

(258) Obligation:

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

QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)

The TRS R consists of the following rules:

le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
QSORT1_IN_AA(x1, x2)  =  QSORT1_IN_AA
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)

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

(259) PiDPToQDPProof (SOUND transformation)

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

(260) Obligation:

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

QSORT1_IN_AA → U24_AA(le36_in_aa)
U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa)
U26_AA(split51_out_aaaa) → QSORT1_IN_AA

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(261) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QSORT1_IN_AA → U24_AA(le36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QSORT1_IN_AA → U24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AA → U24_AA(le36_out_aa(0))

(262) Obligation:

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

U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa)
U26_AA(split51_out_aaaa) → QSORT1_IN_AA
QSORT1_IN_AA → U24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AA → U24_AA(le36_out_aa(0))

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(263) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa)
U24_AA(le36_out_aa(y0)) → U26_AA(U3_aaaa(le36_in_aa))
U24_AA(le36_out_aa(y0)) → U26_AA(U5_aaaa(gt67_in_aa))

(264) Obligation:

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

U26_AA(split51_out_aaaa) → QSORT1_IN_AA
QSORT1_IN_AA → U24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AA → U24_AA(le36_out_aa(0))
U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa)
U24_AA(le36_out_aa(y0)) → U26_AA(U3_aaaa(le36_in_aa))
U24_AA(le36_out_aa(y0)) → U26_AA(U5_aaaa(gt67_in_aa))

The TRS R consists of the following rules:

le36_in_aa → U2_aa(le36_in_aa)
le36_in_aa → le36_out_aa(0)
split51_in_aaaa → split51_out_aaaa
split51_in_aaaa → U3_aaaa(le36_in_aa)
split51_in_aaaa → U5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aa → U7_aa(gt67_in_aa)
gt67_in_aa → gt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(265) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QSORT1_IN_AA evaluates to t =QSORT1_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

QSORT1_IN_AA → U24_AA(le36_out_aa(0))
with rule QSORT1_IN_AA → U24_AA(le36_out_aa(0)) at position [] and matcher [ ]

U24_AA(le36_out_aa(0)) → U26_AA(split51_out_aaaa)
with rule U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa) at position [] and matcher [y0 / 0]

U26_AA(split51_out_aaaa) → QSORT1_IN_AA
with rule U26_AA(split51_out_aaaa) → QSORT1_IN_AA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(266) NO

(267) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)

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

(268) UsableRulesProof (EQUIVALENT transformation)

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

(269) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

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

(270) PiDPToQDPProof (EQUIVALENT transformation)

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

(271) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

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

(272) 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:

• GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
  The graph contains the following edges 1 > 1, 2 > 2

(273) YES

(274) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)

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

(275) UsableRulesProof (EQUIVALENT transformation)

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

(276) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

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

(277) PiDPToQDPProof (EQUIVALENT transformation)

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

(278) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

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

(279) 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:

• LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
  The graph contains the following edges 1 > 1, 2 > 2

(280) YES

(281) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

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

(282) UsableRulesProof (EQUIVALENT transformation)

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

(283) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

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

(284) PiDPToQDPProof (SOUND transformation)

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

(285) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116) → U3_GGAA(T114, T115, T116, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116)
SPLIT51_IN_GGAA(.(T129, T130), T131) → U5_GGAA(T129, T130, T131, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131)

The TRS R consists of the following rules:

le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))

The set Q consists of the following terms:

le36_in_gg(x0, x1)
gt67_in_gg(x0, x1)
U2_gg(x0, x1, x2)
U7_gg(x0, x1, x2)

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

(286) 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:

• U3_GGAA(T114, T115, T116, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116)
  The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2

• U5_GGAA(T129, T130, T131, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131)
  The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2

• SPLIT51_IN_GGAA(.(T114, T115), T116) → U3_GGAA(T114, T115, T116, le36_in_gg(T114, T116))
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

• SPLIT51_IN_GGAA(.(T129, T130), T131) → U5_GGAA(T129, T130, T131, gt67_in_gg(T129, T131))
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

(287) YES

(288) Obligation:

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

APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U1_ggga(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x1, x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x2, x3, x4, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x2, x3, x5)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_out_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x1, x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x1, x2, x3, x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x2, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x1, x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x1, x2, x3, x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)

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

(289) UsableRulesProof (EQUIVALENT transformation)

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

(290) Obligation:

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

APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)

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

(291) PiDPToQDPProof (SOUND transformation)

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

(292) Obligation:

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

APPEND23_IN_GGGA(.(T45, T46), T47, T48) → APPEND23_IN_GGGA(T46, T47, T48)

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

(293) 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:

• APPEND23_IN_GGGA(.(T45, T46), T47, T48) → APPEND23_IN_GGGA(T46, T47, T48)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(294) YES