Left Termination of the query pattern qsort_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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:

Triples:

append23(.(T45, T46), T47, T48, .(T45, T50)) :- append23(T46, T47, T48, T50).
le36(s(T82), s(T83)) :- le36(T82, T83).
split51(.(T114, T115), T116, .(T114, X167), X168) :- le36(T114, T116).
split51(.(T114, T115), T116, .(T114, X167), X168) :- ','(lec36(T114, T116), split51(T115, T116, X167, X168)).
split51(.(T129, T130), T131, X199, .(T129, X200)) :- gt67(T129, T131).
split51(.(T129, T130), T131, X199, .(T129, X200)) :- ','(gtc67(T129, T131), split51(T130, T131, X199, X200)).
gt67(s(T144), s(T145)) :- gt67(T144, T145).
qsort79(.(T163, T164), X265) :- split51(T164, T163, X261, X262).
qsort79(.(T163, T164), X265) :- ','(splitc51(T164, T163, T167, T168), qsort79(T167, X263)).
qsort79(.(T163, T164), X265) :- ','(splitc51(T164, T163, T167, T168), ','(qsortc79(T167, T171), qsort79(T168, X264))).
qsort79(.(T163, T164), X265) :- ','(splitc51(T164, T163, T167, T168), ','(qsortc79(T167, T171), ','(qsortc79(T168, T172), append94(T171, T163, T172, X265)))).
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) :- ','(qsortc79(T94, T158), append23(T155, T69, T158, T9)).
qsort1(.(T14, []), T9) :- qsort14(X12).
qsort1(.(T14, []), T9) :- ','(qsortc14(T17), qsort14(X13)).
qsort1(.(T14, []), T9) :- ','(qsortc14(T17), ','(qsortc14(T20), append23(T17, T14, T20, T9))).
qsort1(.(T69, .(T67, T68)), T9) :- le36(T67, T69).
qsort1(.(T69, .(T67, T68)), T9) :- ','(lec36(T67, T69), split51(T68, T69, X93, X94)).
qsort1(.(T69, .(T67, T68)), T9) :- ','(lec36(T67, T69), ','(splitc51(T68, T69, T93, T94), qsort1(.(T67, T93), X12))).
qsort1(.(T69, .(T67, T68)), T9) :- ','(lec36(T67, T69), ','(splitc51(T68, T69, T93, T94), ','(qsortc1(.(T67, T93), T155), p78(T94, X13, T155, T69, T9)))).
qsort1(.(T215, .(T213, T214)), T9) :- gt67(T213, T215).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gtc67(T213, T215), split51(T214, T215, X323, X324)).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gtc67(T213, T215), ','(splitc51(T214, T215, T222, T223), qsort79(T222, X12))).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gtc67(T213, T215), ','(splitc51(T214, T215, T222, T223), ','(qsortc79(T222, T226), p78(.(T213, T223), X13, T226, T215, T9)))).

Clauses:

qsortc14([]).
appendc23([], T33, T34, .(T33, T34)).
appendc23(.(T45, T46), T47, T48, .(T45, T50)) :- appendc23(T46, T47, T48, T50).
lec36(s(T82), s(T83)) :- lec36(T82, T83).
lec36(0, s(T90)).
lec36(0, 0).
splitc51([], T101, [], []).
splitc51(.(T114, T115), T116, .(T114, X167), X168) :- ','(lec36(T114, T116), splitc51(T115, T116, X167, X168)).
splitc51(.(T129, T130), T131, X199, .(T129, X200)) :- ','(gtc67(T129, T131), splitc51(T130, T131, X199, X200)).
gtc67(s(T144), s(T145)) :- gtc67(T144, T145).
gtc67(s(T150), 0).
qsortc1([], []).
qsortc1(.(T14, []), T9) :- ','(qsortc14(T17), ','(qsortc14(T20), appendc23(T17, T14, T20, T9))).
qsortc1(.(T69, .(T67, T68)), T9) :- ','(lec36(T67, T69), ','(splitc51(T68, T69, T93, T94), ','(qsortc1(.(T67, T93), T155), qc78(T94, X13, T155, T69, T9)))).
qsortc1(.(T215, .(T213, T214)), T9) :- ','(gtc67(T213, T215), ','(splitc51(T214, T215, T222, T223), ','(qsortc79(T222, T226), qc78(.(T213, T223), X13, T226, T215, T9)))).
qsortc79([], []).
qsortc79(.(T163, T164), X265) :- ','(splitc51(T164, T163, T167, T168), ','(qsortc79(T167, T171), ','(qsortc79(T168, T172), appendc94(T171, T163, T172, X265)))).
appendc94([], T185, T186, .(T185, T186)).
appendc94(.(T195, T196), T197, T198, .(T195, X295)) :- appendc94(T196, T197, T198, X295).
qc78(T94, T158, T155, T69, T9) :- ','(qsortc79(T94, T158), appendc23(T155, T69, T158, T9)).

Afs:

qsort1(x1, x2)  =  qsort1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

append23(.(T45, T46), T47, T48, .(T45, T50)) :- append23(T46, T47, T48, T50).
le36(s(T82), s(T83)) :- le36(T82, T83).
split51(.(T114, T115), T116, .(T114, X167), X168) :- le36(T114, T116).
split51(.(T114, T115), T116, .(T114, X167), X168) :- ','(lec36(T114, T116), split51(T115, T116, X167, X168)).
split51(.(T129, T130), T131, X199, .(T129, X200)) :- gt67(T129, T131).
split51(.(T129, T130), T131, X199, .(T129, X200)) :- ','(gtc67(T129, T131), split51(T130, T131, X199, X200)).
gt67(s(T144), s(T145)) :- gt67(T144, T145).
qsort79(.(T163, T164), X265) :- split51(T164, T163, X261, X262).
qsort79(.(T163, T164), X265) :- ','(splitc51(T164, T163, T167, T168), qsort79(T167, X263)).
qsort79(.(T163, T164), X265) :- ','(splitc51(T164, T163, T167, T168), ','(qsortc79(T167, T171), qsort79(T168, X264))).
qsort79(.(T163, T164), X265) :- ','(splitc51(T164, T163, T167, T168), ','(qsortc79(T167, T171), ','(qsortc79(T168, T172), append94(T171, T163, T172, X265)))).
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) :- ','(qsortc79(T94, T158), append23(T155, T69, T158, T9)).
qsort1(.(T14, []), T9) :- ','(qsortc14(T17), ','(qsortc14(T20), append23(T17, T14, T20, T9))).
qsort1(.(T69, .(T67, T68)), T9) :- le36(T67, T69).
qsort1(.(T69, .(T67, T68)), T9) :- ','(lec36(T67, T69), split51(T68, T69, X93, X94)).
qsort1(.(T69, .(T67, T68)), T9) :- ','(lec36(T67, T69), ','(splitc51(T68, T69, T93, T94), qsort1(.(T67, T93), X12))).
qsort1(.(T69, .(T67, T68)), T9) :- ','(lec36(T67, T69), ','(splitc51(T68, T69, T93, T94), ','(qsortc1(.(T67, T93), T155), p78(T94, X13, T155, T69, T9)))).
qsort1(.(T215, .(T213, T214)), T9) :- gt67(T213, T215).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gtc67(T213, T215), split51(T214, T215, X323, X324)).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gtc67(T213, T215), ','(splitc51(T214, T215, T222, T223), qsort79(T222, X12))).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gtc67(T213, T215), ','(splitc51(T214, T215, T222, T223), ','(qsortc79(T222, T226), p78(.(T213, T223), X13, T226, T215, T9)))).

Clauses:

qsortc14([]).
appendc23([], T33, T34, .(T33, T34)).
appendc23(.(T45, T46), T47, T48, .(T45, T50)) :- appendc23(T46, T47, T48, T50).
lec36(s(T82), s(T83)) :- lec36(T82, T83).
lec36(0, s(T90)).
lec36(0, 0).
splitc51([], T101, [], []).
splitc51(.(T114, T115), T116, .(T114, X167), X168) :- ','(lec36(T114, T116), splitc51(T115, T116, X167, X168)).
splitc51(.(T129, T130), T131, X199, .(T129, X200)) :- ','(gtc67(T129, T131), splitc51(T130, T131, X199, X200)).
gtc67(s(T144), s(T145)) :- gtc67(T144, T145).
gtc67(s(T150), 0).
qsortc1([], []).
qsortc1(.(T14, []), T9) :- ','(qsortc14(T17), ','(qsortc14(T20), appendc23(T17, T14, T20, T9))).
qsortc1(.(T69, .(T67, T68)), T9) :- ','(lec36(T67, T69), ','(splitc51(T68, T69, T93, T94), ','(qsortc1(.(T67, T93), T155), qc78(T94, X13, T155, T69, T9)))).
qsortc1(.(T215, .(T213, T214)), T9) :- ','(gtc67(T213, T215), ','(splitc51(T214, T215, T222, T223), ','(qsortc79(T222, T226), qc78(.(T213, T223), X13, T226, T215, T9)))).
qsortc79([], []).
qsortc79(.(T163, T164), X265) :- ','(splitc51(T164, T163, T167, T168), ','(qsortc79(T167, T171), ','(qsortc79(T168, T172), appendc94(T171, T163, T172, X265)))).
appendc94([], T185, T186, .(T185, T186)).
appendc94(.(T195, T196), T197, T198, .(T195, X295)) :- appendc94(T196, T197, T198, X295).
qc78(T94, T158, T155, T69, T9) :- ','(qsortc79(T94, T158), appendc23(T155, T69, T158, T9)).

Afs:

qsort1(x1, x2)  =  qsort1(x1)

(5) TriplesToPiDPProof (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)
append23_in: (b,b,b,f)
le36_in: (b,b)
lec36_in: (b,b)
split51_in: (b,b,f,f)
gt67_in: (b,b)
gtc67_in: (b,b)
splitc51_in: (b,b,f,f)
qsort79_in: (b,f)
qsortc79_in: (b,f)
appendc94_in: (b,b,b,f)
append94_in: (b,b,b,f)
p78_in: (b,f,b,b,f)
qsortc1_in: (b,f)
appendc23_in: (b,b,b,f)
qc78_in: (b,f,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

QSORT1_IN_GA(.(T14, []), T9) → U21_GA(T14, T9, qsortc14_in_a(T17))
U21_GA(T14, T9, qsortc14_out_a(T17)) → U22_GA(T14, T9, T17, qsortc14_in_a(T20))
U22_GA(T14, T9, T17, qsortc14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsortc14_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)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U25_GA(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U25_GA(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U25_GA(T69, T67, T68, T9, lec36_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)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U4_GGAA(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U4_GGAA(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U5_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U4_GGAA(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U6_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)) → U9_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U7_GGAA(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U7_GGAA(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U8_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U7_GGAA(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U25_GA(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U27_GA(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, qsort1_in_ga(.(T67, T93), X12))
U27_GA(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_GA(.(T67, T93), X12)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U31_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U32_GA(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U32_GA(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U33_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U32_GA(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U32_GA(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U34_GA(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U34_GA(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U35_GA(T215, T213, T214, T9, qsort79_in_ga(T222, X12))
U34_GA(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_GA(T222, X12)
QSORT79_IN_GA(.(T163, T164), X265) → U10_GA(T163, T164, X265, split51_in_ggaa(T164, T163, X261, X262))
QSORT79_IN_GA(.(T163, T164), X265) → SPLIT51_IN_GGAA(T164, T163, X261, X262)
QSORT79_IN_GA(.(T163, T164), X265) → U11_GA(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U12_GA(T163, T164, X265, qsort79_in_ga(T167, X263))
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → QSORT79_IN_GA(T167, X263)
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U13_GA(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U13_GA(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U14_GA(T163, T164, X265, qsort79_in_ga(T168, X264))
U13_GA(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → QSORT79_IN_GA(T168, X264)
U13_GA(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U15_GA(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U15_GA(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U16_GA(T163, T164, X265, append94_in_ggga(T171, T163, T172, X265))
U15_GA(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → APPEND94_IN_GGGA(T171, T163, T172, X265)
APPEND94_IN_GGGA(.(T195, T196), T197, T198, .(T195, X295)) → U17_GGGA(T195, T196, T197, T198, X295, append94_in_ggga(T196, T197, T198, X295))
APPEND94_IN_GGGA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_GGGA(T196, T197, T198, X295)
U34_GA(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U36_GA(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U36_GA(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U37_GA(T215, T213, T214, T9, p78_in_gagga(.(T213, T223), X13, T226, T215, T9))
U36_GA(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → P78_IN_GAGGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_GAGGA(T94, X13, T155, T69, T9) → U18_GAGGA(T94, X13, T155, T69, T9, qsort79_in_ga(T94, X13))
P78_IN_GAGGA(T94, X13, T155, T69, T9) → QSORT79_IN_GA(T94, X13)
P78_IN_GAGGA(T94, T158, T155, T69, T9) → U19_GAGGA(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U19_GAGGA(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U20_GAGGA(T94, T158, T155, T69, T9, append23_in_ggga(T155, T69, T158, T9))
U19_GAGGA(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → APPEND23_IN_GGGA(T155, T69, T158, T9)
U27_GA(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U29_GA(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
U29_GA(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U30_GA(T69, T67, T68, T9, p78_in_gagga(T94, X13, T155, T69, T9))
U29_GA(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → P78_IN_GAGGA(T94, X13, T155, T69, T9)

The TRS R consists of the following rules:

qsortc14_in_a([]) → qsortc14_out_a([])
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
qsortc1_in_ga([], []) → qsortc1_out_ga([], [])
qsortc1_in_ga(.(T14, []), T9) → U46_ga(T14, T9, qsortc14_in_a(T17))
U46_ga(T14, T9, qsortc14_out_a(T17)) → U47_ga(T14, T9, T17, qsortc14_in_a(T20))
U47_ga(T14, T9, T17, qsortc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, .(T33, T34)) → appendc23_out_ggga([], T33, T34, .(T33, T34))
appendc23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsortc1_out_ga(.(T14, []), T9)
qsortc1_in_ga(.(T69, .(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U50_ga(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U51_ga(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
qsortc1_in_ga(.(T215, .(T213, T214)), T9) → U53_ga(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U53_ga(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U54_ga(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U54_ga(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U55_ga(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U55_ga(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U56_ga(T215, T213, T214, T9, qc78_in_gagga(.(T213, T223), X13, T226, T215, T9))
qc78_in_gagga(T94, T158, T155, T69, T9) → U62_gagga(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U62_gagga(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U63_gagga(T94, T158, T155, T69, T9, appendc23_in_ggga(T155, T69, T158, T9))
U63_gagga(T94, T158, T155, T69, T9, appendc23_out_ggga(T155, T69, T158, T9)) → qc78_out_gagga(T94, T158, T155, T69, T9)
U56_ga(T215, T213, T214, T9, qc78_out_gagga(.(T213, T223), X13, T226, T215, T9)) → qsortc1_out_ga(.(T215, .(T213, T214)), T9)
U51_ga(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T94, X13, T155, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T94, X13, T155, T69, T9)) → qsortc1_out_ga(.(T69, .(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
qsortc14_in_a(x1)  =  qsortc14_in_a
qsortc14_out_a(x1)  =  qsortc14_out_a(x1)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsort79_in_ga(x1, x2)  =  qsort79_in_ga(x1)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
append94_in_ggga(x1, x2, x3, x4)  =  append94_in_ggga(x1, x2, x3)
p78_in_gagga(x1, x2, x3, x4, x5)  =  p78_in_gagga(x1, x3, x4)
qsortc1_in_ga(x1, x2)  =  qsortc1_in_ga(x1)
qsortc1_out_ga(x1, x2)  =  qsortc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
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)
U26_GA(x1, x2, x3, x4, x5)  =  U26_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)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x2, x3, x6)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3, x4, x5)  =  U28_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)  =  U34_GA(x1, x2, x3, x5)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x1, x2, x3, x5)
QSORT79_IN_GA(x1, x2)  =  QSORT79_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x4, x5)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x2, x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x4, x5)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x2, x4)
APPEND94_IN_GGGA(x1, x2, x3, x4)  =  APPEND94_IN_GGGA(x1, x2, x3)
U17_GGGA(x1, x2, x3, x4, x5, x6)  =  U17_GGGA(x1, x2, x3, x4, x6)
U36_GA(x1, x2, x3, x4, x5, x6)  =  U36_GA(x1, x2, x3, x5, x6)
U37_GA(x1, x2, x3, x4, x5)  =  U37_GA(x1, x2, x3, x5)
P78_IN_GAGGA(x1, x2, x3, x4, x5)  =  P78_IN_GAGGA(x1, x3, x4)
U18_GAGGA(x1, x2, x3, x4, x5, x6)  =  U18_GAGGA(x1, x3, x4, x6)
U19_GAGGA(x1, x2, x3, x4, x5, x6)  =  U19_GAGGA(x1, x3, x4, x6)
U20_GAGGA(x1, x2, x3, x4, x5, x6)  =  U20_GAGGA(x1, x3, x4, x6)
U29_GA(x1, x2, x3, x4, x5, x6)  =  U29_GA(x1, x2, x3, x5, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x2, x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

QSORT1_IN_GA(.(T14, []), T9) → U21_GA(T14, T9, qsortc14_in_a(T17))
U21_GA(T14, T9, qsortc14_out_a(T17)) → U22_GA(T14, T9, T17, qsortc14_in_a(T20))
U22_GA(T14, T9, T17, qsortc14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsortc14_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)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U25_GA(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U25_GA(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U25_GA(T69, T67, T68, T9, lec36_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)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U4_GGAA(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U4_GGAA(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U5_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U4_GGAA(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U6_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)) → U9_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U7_GGAA(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U7_GGAA(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U8_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U7_GGAA(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U25_GA(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U27_GA(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, qsort1_in_ga(.(T67, T93), X12))
U27_GA(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_GA(.(T67, T93), X12)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U31_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U32_GA(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U32_GA(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U33_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U32_GA(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U32_GA(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U34_GA(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U34_GA(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U35_GA(T215, T213, T214, T9, qsort79_in_ga(T222, X12))
U34_GA(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_GA(T222, X12)
QSORT79_IN_GA(.(T163, T164), X265) → U10_GA(T163, T164, X265, split51_in_ggaa(T164, T163, X261, X262))
QSORT79_IN_GA(.(T163, T164), X265) → SPLIT51_IN_GGAA(T164, T163, X261, X262)
QSORT79_IN_GA(.(T163, T164), X265) → U11_GA(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U12_GA(T163, T164, X265, qsort79_in_ga(T167, X263))
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → QSORT79_IN_GA(T167, X263)
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U13_GA(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U13_GA(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U14_GA(T163, T164, X265, qsort79_in_ga(T168, X264))
U13_GA(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → QSORT79_IN_GA(T168, X264)
U13_GA(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U15_GA(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U15_GA(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U16_GA(T163, T164, X265, append94_in_ggga(T171, T163, T172, X265))
U15_GA(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → APPEND94_IN_GGGA(T171, T163, T172, X265)
APPEND94_IN_GGGA(.(T195, T196), T197, T198, .(T195, X295)) → U17_GGGA(T195, T196, T197, T198, X295, append94_in_ggga(T196, T197, T198, X295))
APPEND94_IN_GGGA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_GGGA(T196, T197, T198, X295)
U34_GA(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U36_GA(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U36_GA(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U37_GA(T215, T213, T214, T9, p78_in_gagga(.(T213, T223), X13, T226, T215, T9))
U36_GA(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → P78_IN_GAGGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_GAGGA(T94, X13, T155, T69, T9) → U18_GAGGA(T94, X13, T155, T69, T9, qsort79_in_ga(T94, X13))
P78_IN_GAGGA(T94, X13, T155, T69, T9) → QSORT79_IN_GA(T94, X13)
P78_IN_GAGGA(T94, T158, T155, T69, T9) → U19_GAGGA(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U19_GAGGA(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U20_GAGGA(T94, T158, T155, T69, T9, append23_in_ggga(T155, T69, T158, T9))
U19_GAGGA(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → APPEND23_IN_GGGA(T155, T69, T158, T9)
U27_GA(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U29_GA(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
U29_GA(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U30_GA(T69, T67, T68, T9, p78_in_gagga(T94, X13, T155, T69, T9))
U29_GA(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → P78_IN_GAGGA(T94, X13, T155, T69, T9)

The TRS R consists of the following rules:

qsortc14_in_a([]) → qsortc14_out_a([])
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
qsortc1_in_ga([], []) → qsortc1_out_ga([], [])
qsortc1_in_ga(.(T14, []), T9) → U46_ga(T14, T9, qsortc14_in_a(T17))
U46_ga(T14, T9, qsortc14_out_a(T17)) → U47_ga(T14, T9, T17, qsortc14_in_a(T20))
U47_ga(T14, T9, T17, qsortc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, .(T33, T34)) → appendc23_out_ggga([], T33, T34, .(T33, T34))
appendc23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsortc1_out_ga(.(T14, []), T9)
qsortc1_in_ga(.(T69, .(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U50_ga(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U51_ga(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
qsortc1_in_ga(.(T215, .(T213, T214)), T9) → U53_ga(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U53_ga(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U54_ga(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U54_ga(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U55_ga(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U55_ga(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U56_ga(T215, T213, T214, T9, qc78_in_gagga(.(T213, T223), X13, T226, T215, T9))
qc78_in_gagga(T94, T158, T155, T69, T9) → U62_gagga(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U62_gagga(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U63_gagga(T94, T158, T155, T69, T9, appendc23_in_ggga(T155, T69, T158, T9))
U63_gagga(T94, T158, T155, T69, T9, appendc23_out_ggga(T155, T69, T158, T9)) → qc78_out_gagga(T94, T158, T155, T69, T9)
U56_ga(T215, T213, T214, T9, qc78_out_gagga(.(T213, T223), X13, T226, T215, T9)) → qsortc1_out_ga(.(T215, .(T213, T214)), T9)
U51_ga(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T94, X13, T155, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T94, X13, T155, T69, T9)) → qsortc1_out_ga(.(T69, .(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
qsortc14_in_a(x1)  =  qsortc14_in_a
qsortc14_out_a(x1)  =  qsortc14_out_a(x1)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsort79_in_ga(x1, x2)  =  qsort79_in_ga(x1)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
append94_in_ggga(x1, x2, x3, x4)  =  append94_in_ggga(x1, x2, x3)
p78_in_gagga(x1, x2, x3, x4, x5)  =  p78_in_gagga(x1, x3, x4)
qsortc1_in_ga(x1, x2)  =  qsortc1_in_ga(x1)
qsortc1_out_ga(x1, x2)  =  qsortc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
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)
U26_GA(x1, x2, x3, x4, x5)  =  U26_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)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x2, x3, x6)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3, x4, x5)  =  U28_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)  =  U34_GA(x1, x2, x3, x5)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x1, x2, x3, x5)
QSORT79_IN_GA(x1, x2)  =  QSORT79_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x4, x5)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x2, x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x4, x5)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x2, x4)
APPEND94_IN_GGGA(x1, x2, x3, x4)  =  APPEND94_IN_GGGA(x1, x2, x3)
U17_GGGA(x1, x2, x3, x4, x5, x6)  =  U17_GGGA(x1, x2, x3, x4, x6)
U36_GA(x1, x2, x3, x4, x5, x6)  =  U36_GA(x1, x2, x3, x5, x6)
U37_GA(x1, x2, x3, x4, x5)  =  U37_GA(x1, x2, x3, x5)
P78_IN_GAGGA(x1, x2, x3, x4, x5)  =  P78_IN_GAGGA(x1, x3, x4)
U18_GAGGA(x1, x2, x3, x4, x5, x6)  =  U18_GAGGA(x1, x3, x4, x6)
U19_GAGGA(x1, x2, x3, x4, x5, x6)  =  U19_GAGGA(x1, x3, x4, x6)
U20_GAGGA(x1, x2, x3, x4, x5, x6)  =  U20_GAGGA(x1, x3, x4, x6)
U29_GA(x1, x2, x3, x4, x5, x6)  =  U29_GA(x1, x2, x3, x5, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x2, x3, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 45 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND94_IN_GGGA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_GGGA(T196, T197, T198, X295)

The TRS R consists of the following rules:

qsortc14_in_a([]) → qsortc14_out_a([])
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
qsortc1_in_ga([], []) → qsortc1_out_ga([], [])
qsortc1_in_ga(.(T14, []), T9) → U46_ga(T14, T9, qsortc14_in_a(T17))
U46_ga(T14, T9, qsortc14_out_a(T17)) → U47_ga(T14, T9, T17, qsortc14_in_a(T20))
U47_ga(T14, T9, T17, qsortc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, .(T33, T34)) → appendc23_out_ggga([], T33, T34, .(T33, T34))
appendc23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsortc1_out_ga(.(T14, []), T9)
qsortc1_in_ga(.(T69, .(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U50_ga(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U51_ga(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
qsortc1_in_ga(.(T215, .(T213, T214)), T9) → U53_ga(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U53_ga(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U54_ga(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U54_ga(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U55_ga(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U55_ga(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U56_ga(T215, T213, T214, T9, qc78_in_gagga(.(T213, T223), X13, T226, T215, T9))
qc78_in_gagga(T94, T158, T155, T69, T9) → U62_gagga(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U62_gagga(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U63_gagga(T94, T158, T155, T69, T9, appendc23_in_ggga(T155, T69, T158, T9))
U63_gagga(T94, T158, T155, T69, T9, appendc23_out_ggga(T155, T69, T158, T9)) → qc78_out_gagga(T94, T158, T155, T69, T9)
U56_ga(T215, T213, T214, T9, qc78_out_gagga(.(T213, T223), X13, T226, T215, T9)) → qsortc1_out_ga(.(T215, .(T213, T214)), T9)
U51_ga(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T94, X13, T155, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T94, X13, T155, T69, T9)) → qsortc1_out_ga(.(T69, .(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
qsortc14_in_a(x1)  =  qsortc14_in_a
qsortc14_out_a(x1)  =  qsortc14_out_a(x1)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsortc1_in_ga(x1, x2)  =  qsortc1_in_ga(x1)
qsortc1_out_ga(x1, x2)  =  qsortc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
APPEND94_IN_GGGA(x1, x2, x3, x4)  =  APPEND94_IN_GGGA(x1, x2, x3)

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:

APPEND94_IN_GGGA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_GGGA(T196, T197, T198, X295)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND94_IN_GGGA(x1, x2, x3, x4)  =  APPEND94_IN_GGGA(x1, x2, x3)

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:

APPEND94_IN_GGGA(.(T195, T196), T197, T198) → APPEND94_IN_GGGA(T196, T197, T198)

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

(14) 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:

• APPEND94_IN_GGGA(.(T195, T196), T197, T198) → APPEND94_IN_GGGA(T196, T197, T198)
  The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(15) YES

(16) 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:

qsortc14_in_a([]) → qsortc14_out_a([])
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
qsortc1_in_ga([], []) → qsortc1_out_ga([], [])
qsortc1_in_ga(.(T14, []), T9) → U46_ga(T14, T9, qsortc14_in_a(T17))
U46_ga(T14, T9, qsortc14_out_a(T17)) → U47_ga(T14, T9, T17, qsortc14_in_a(T20))
U47_ga(T14, T9, T17, qsortc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, .(T33, T34)) → appendc23_out_ggga([], T33, T34, .(T33, T34))
appendc23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsortc1_out_ga(.(T14, []), T9)
qsortc1_in_ga(.(T69, .(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U50_ga(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U51_ga(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
qsortc1_in_ga(.(T215, .(T213, T214)), T9) → U53_ga(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U53_ga(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U54_ga(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U54_ga(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U55_ga(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U55_ga(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U56_ga(T215, T213, T214, T9, qc78_in_gagga(.(T213, T223), X13, T226, T215, T9))
qc78_in_gagga(T94, T158, T155, T69, T9) → U62_gagga(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U62_gagga(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U63_gagga(T94, T158, T155, T69, T9, appendc23_in_ggga(T155, T69, T158, T9))
U63_gagga(T94, T158, T155, T69, T9, appendc23_out_ggga(T155, T69, T158, T9)) → qc78_out_gagga(T94, T158, T155, T69, T9)
U56_ga(T215, T213, T214, T9, qc78_out_gagga(.(T213, T223), X13, T226, T215, T9)) → qsortc1_out_ga(.(T215, .(T213, T214)), T9)
U51_ga(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T94, X13, T155, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T94, X13, T155, T69, T9)) → qsortc1_out_ga(.(T69, .(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
qsortc14_in_a(x1)  =  qsortc14_in_a
qsortc14_out_a(x1)  =  qsortc14_out_a(x1)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsortc1_in_ga(x1, x2)  =  qsortc1_in_ga(x1)
qsortc1_out_ga(x1, x2)  =  qsortc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, 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:

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

(19) PiDPToQDPProof (EQUIVALENT 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:

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.

(21) 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

(22) YES

(23) 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:

qsortc14_in_a([]) → qsortc14_out_a([])
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
qsortc1_in_ga([], []) → qsortc1_out_ga([], [])
qsortc1_in_ga(.(T14, []), T9) → U46_ga(T14, T9, qsortc14_in_a(T17))
U46_ga(T14, T9, qsortc14_out_a(T17)) → U47_ga(T14, T9, T17, qsortc14_in_a(T20))
U47_ga(T14, T9, T17, qsortc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, .(T33, T34)) → appendc23_out_ggga([], T33, T34, .(T33, T34))
appendc23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsortc1_out_ga(.(T14, []), T9)
qsortc1_in_ga(.(T69, .(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U50_ga(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U51_ga(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
qsortc1_in_ga(.(T215, .(T213, T214)), T9) → U53_ga(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U53_ga(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U54_ga(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U54_ga(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U55_ga(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U55_ga(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U56_ga(T215, T213, T214, T9, qc78_in_gagga(.(T213, T223), X13, T226, T215, T9))
qc78_in_gagga(T94, T158, T155, T69, T9) → U62_gagga(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U62_gagga(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U63_gagga(T94, T158, T155, T69, T9, appendc23_in_ggga(T155, T69, T158, T9))
U63_gagga(T94, T158, T155, T69, T9, appendc23_out_ggga(T155, T69, T158, T9)) → qc78_out_gagga(T94, T158, T155, T69, T9)
U56_ga(T215, T213, T214, T9, qc78_out_gagga(.(T213, T223), X13, T226, T215, T9)) → qsortc1_out_ga(.(T215, .(T213, T214)), T9)
U51_ga(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T94, X13, T155, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T94, X13, T155, T69, T9)) → qsortc1_out_ga(.(T69, .(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
qsortc14_in_a(x1)  =  qsortc14_in_a
qsortc14_out_a(x1)  =  qsortc14_out_a(x1)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsortc1_in_ga(x1, x2)  =  qsortc1_in_ga(x1)
qsortc1_out_ga(x1, x2)  =  qsortc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)

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:

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

(26) PiDPToQDPProof (EQUIVALENT 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:

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.

(28) 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

(29) YES

(30) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U4_GGAA(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U4_GGAA(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U7_GGAA(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U7_GGAA(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

qsortc14_in_a([]) → qsortc14_out_a([])
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
qsortc1_in_ga([], []) → qsortc1_out_ga([], [])
qsortc1_in_ga(.(T14, []), T9) → U46_ga(T14, T9, qsortc14_in_a(T17))
U46_ga(T14, T9, qsortc14_out_a(T17)) → U47_ga(T14, T9, T17, qsortc14_in_a(T20))
U47_ga(T14, T9, T17, qsortc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, .(T33, T34)) → appendc23_out_ggga([], T33, T34, .(T33, T34))
appendc23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsortc1_out_ga(.(T14, []), T9)
qsortc1_in_ga(.(T69, .(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U50_ga(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U51_ga(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
qsortc1_in_ga(.(T215, .(T213, T214)), T9) → U53_ga(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U53_ga(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U54_ga(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U54_ga(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U55_ga(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U55_ga(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U56_ga(T215, T213, T214, T9, qc78_in_gagga(.(T213, T223), X13, T226, T215, T9))
qc78_in_gagga(T94, T158, T155, T69, T9) → U62_gagga(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U62_gagga(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U63_gagga(T94, T158, T155, T69, T9, appendc23_in_ggga(T155, T69, T158, T9))
U63_gagga(T94, T158, T155, T69, T9, appendc23_out_ggga(T155, T69, T158, T9)) → qc78_out_gagga(T94, T158, T155, T69, T9)
U56_ga(T215, T213, T214, T9, qc78_out_gagga(.(T213, T223), X13, T226, T215, T9)) → qsortc1_out_ga(.(T215, .(T213, T214)), T9)
U51_ga(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T94, X13, T155, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T94, X13, T155, T69, T9)) → qsortc1_out_ga(.(T69, .(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
qsortc14_in_a(x1)  =  qsortc14_in_a
qsortc14_out_a(x1)  =  qsortc14_out_a(x1)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsortc1_in_ga(x1, x2)  =  qsortc1_in_ga(x1)
qsortc1_out_ga(x1, x2)  =  qsortc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

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:

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U4_GGAA(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U4_GGAA(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U7_GGAA(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U7_GGAA(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

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:

SPLIT51_IN_GGAA(.(T114, T115), T116) → U4_GGAA(T114, T115, T116, lec36_in_gg(T114, T116))
U4_GGAA(T114, T115, T116, lec36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116)
SPLIT51_IN_GGAA(.(T129, T130), T131) → U7_GGAA(T129, T130, T131, gtc67_in_gg(T129, T131))
U7_GGAA(T129, T130, T131, gtc67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131)

The TRS R consists of the following rules:

lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))

The set Q consists of the following terms:

lec36_in_gg(x0, x1)
gtc67_in_gg(x0, x1)
U40_gg(x0, x1, x2)
U45_gg(x0, x1, x2)

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:

• U4_GGAA(T114, T115, T116, lec36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116)
  The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2

• U7_GGAA(T129, T130, T131, gtc67_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) → U4_GGAA(T114, T115, T116, lec36_in_gg(T114, T116))
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

• SPLIT51_IN_GGAA(.(T129, T130), T131) → U7_GGAA(T129, T130, T131, gtc67_in_gg(T129, T131))
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

(36) YES

(37) Obligation:

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

QSORT79_IN_GA(.(T163, T164), X265) → U11_GA(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → QSORT79_IN_GA(T167, X263)
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U13_GA(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U13_GA(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → QSORT79_IN_GA(T168, X264)

The TRS R consists of the following rules:

qsortc14_in_a([]) → qsortc14_out_a([])
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
qsortc1_in_ga([], []) → qsortc1_out_ga([], [])
qsortc1_in_ga(.(T14, []), T9) → U46_ga(T14, T9, qsortc14_in_a(T17))
U46_ga(T14, T9, qsortc14_out_a(T17)) → U47_ga(T14, T9, T17, qsortc14_in_a(T20))
U47_ga(T14, T9, T17, qsortc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, .(T33, T34)) → appendc23_out_ggga([], T33, T34, .(T33, T34))
appendc23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsortc1_out_ga(.(T14, []), T9)
qsortc1_in_ga(.(T69, .(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U50_ga(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U51_ga(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
qsortc1_in_ga(.(T215, .(T213, T214)), T9) → U53_ga(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U53_ga(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U54_ga(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U54_ga(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U55_ga(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U55_ga(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U56_ga(T215, T213, T214, T9, qc78_in_gagga(.(T213, T223), X13, T226, T215, T9))
qc78_in_gagga(T94, T158, T155, T69, T9) → U62_gagga(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U62_gagga(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U63_gagga(T94, T158, T155, T69, T9, appendc23_in_ggga(T155, T69, T158, T9))
U63_gagga(T94, T158, T155, T69, T9, appendc23_out_ggga(T155, T69, T158, T9)) → qc78_out_gagga(T94, T158, T155, T69, T9)
U56_ga(T215, T213, T214, T9, qc78_out_gagga(.(T213, T223), X13, T226, T215, T9)) → qsortc1_out_ga(.(T215, .(T213, T214)), T9)
U51_ga(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T94, X13, T155, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T94, X13, T155, T69, T9)) → qsortc1_out_ga(.(T69, .(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
qsortc14_in_a(x1)  =  qsortc14_in_a
qsortc14_out_a(x1)  =  qsortc14_out_a(x1)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsortc1_in_ga(x1, x2)  =  qsortc1_in_ga(x1)
qsortc1_out_ga(x1, x2)  =  qsortc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
QSORT79_IN_GA(x1, x2)  =  QSORT79_IN_GA(x1)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x4, x5)

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:

QSORT79_IN_GA(.(T163, T164), X265) → U11_GA(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → QSORT79_IN_GA(T167, X263)
U11_GA(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U13_GA(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U13_GA(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → QSORT79_IN_GA(T168, X264)

The TRS R consists of the following rules:

splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
QSORT79_IN_GA(x1, x2)  =  QSORT79_IN_GA(x1)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x4, x5)

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:

QSORT79_IN_GA(.(T163, T164)) → U11_GA(T163, T164, splitc51_in_ggaa(T164, T163))
U11_GA(T163, T164, splitc51_out_ggaa(T164, T163, T167, T168)) → QSORT79_IN_GA(T167)
U11_GA(T163, T164, splitc51_out_ggaa(T164, T163, T167, T168)) → U13_GA(T163, T164, T168, qsortc79_in_ga(T167))
U13_GA(T163, T164, T168, qsortc79_out_ga(T167, T171)) → QSORT79_IN_GA(T168)

The TRS R consists of the following rules:

splitc51_in_ggaa([], T101) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116) → U41_ggaa(T114, T115, T116, lec36_in_gg(T114, T116))
splitc51_in_ggaa(.(T129, T130), T131) → U43_ggaa(T129, T130, T131, gtc67_in_gg(T129, T131))
qsortc79_in_ga([]) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164)) → U57_ga(T163, T164, splitc51_in_ggaa(T164, T163))
U41_ggaa(T114, T115, T116, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, splitc51_in_ggaa(T115, T116))
U43_ggaa(T129, T130, T131, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, splitc51_in_ggaa(T130, T131))
U57_ga(T163, T164, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, T168, qsortc79_in_ga(T167))
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U42_ggaa(T114, T115, T116, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U44_ggaa(T129, T130, T131, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U58_ga(T163, T164, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, T171, qsortc79_in_ga(T168))
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
U59_ga(T163, T164, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, appendc94_in_ggga(T171, T163, T172))
U60_ga(T163, T164, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
appendc94_in_ggga([], T185, T186) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198) → U61_ggga(T195, T196, T197, T198, appendc94_in_ggga(T196, T197, T198))
U61_ggga(T195, T196, T197, T198, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))

The set Q consists of the following terms:

splitc51_in_ggaa(x0, x1)
qsortc79_in_ga(x0)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U57_ga(x0, x1, x2)
lec36_in_gg(x0, x1)
U42_ggaa(x0, x1, x2, x3)
gtc67_in_gg(x0, x1)
U44_ggaa(x0, x1, x2, x3)
U58_ga(x0, x1, x2, x3)
U40_gg(x0, x1, x2)
U45_gg(x0, x1, x2)
U59_ga(x0, x1, x2, x3)
U60_ga(x0, x1, x2)
appendc94_in_ggga(x0, x1, x2)
U61_ggga(x0, x1, x2, x3, x4)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QSORT79_IN_GA(.(T163, T164)) → U11_GA(T163, T164, splitc51_in_ggaa(T164, T163))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(0) = 0   
POL(QSORT79_IN_GA(x₁)) = x₁   
POL(U11_GA(x₁, x₂, x₃)) = x₃   
POL(U13_GA(x₁, x₂, x₃, x₄)) = x₃   
POL(U40_gg(x₁, x₂, x₃)) = 0   
POL(U41_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U42_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U43_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U44_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U45_gg(x₁, x₂, x₃)) = 0   
POL(U57_ga(x₁, x₂, x₃)) = 0   
POL(U58_ga(x₁, x₂, x₃, x₄)) = 0   
POL(U59_ga(x₁, x₂, x₃, x₄)) = 0   
POL(U60_ga(x₁, x₂, x₃)) = 0   
POL(U61_ggga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL([]) = 0   
POL(appendc94_in_ggga(x₁, x₂, x₃)) = 0   
POL(appendc94_out_ggga(x₁, x₂, x₃, x₄)) = 0   
POL(gtc67_in_gg(x₁, x₂)) = 0   
POL(gtc67_out_gg(x₁, x₂)) = 0   
POL(lec36_in_gg(x₁, x₂)) = 0   
POL(lec36_out_gg(x₁, x₂)) = 0   
POL(qsortc79_in_ga(x₁)) = 0   
POL(qsortc79_out_ga(x₁, x₂)) = 0   
POL(s(x₁)) = 0   
POL(splitc51_in_ggaa(x₁, x₂)) = x₁   
POL(splitc51_out_ggaa(x₁, x₂, x₃, x₄)) = x₃ + x₄   

The following usable rules [FROCOS05] were oriented:

splitc51_in_ggaa([], T101) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116) → U41_ggaa(T114, T115, T116, lec36_in_gg(T114, T116))
splitc51_in_ggaa(.(T129, T130), T131) → U43_ggaa(T129, T130, T131, gtc67_in_gg(T129, T131))
U41_ggaa(T114, T115, T116, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, splitc51_in_ggaa(T115, T116))
U42_ggaa(T114, T115, T116, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U43_ggaa(T129, T130, T131, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, splitc51_in_ggaa(T130, T131))
U44_ggaa(T129, T130, T131, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))

(43) Obligation:

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

U11_GA(T163, T164, splitc51_out_ggaa(T164, T163, T167, T168)) → QSORT79_IN_GA(T167)
U11_GA(T163, T164, splitc51_out_ggaa(T164, T163, T167, T168)) → U13_GA(T163, T164, T168, qsortc79_in_ga(T167))
U13_GA(T163, T164, T168, qsortc79_out_ga(T167, T171)) → QSORT79_IN_GA(T168)

The TRS R consists of the following rules:

splitc51_in_ggaa([], T101) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116) → U41_ggaa(T114, T115, T116, lec36_in_gg(T114, T116))
splitc51_in_ggaa(.(T129, T130), T131) → U43_ggaa(T129, T130, T131, gtc67_in_gg(T129, T131))
qsortc79_in_ga([]) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164)) → U57_ga(T163, T164, splitc51_in_ggaa(T164, T163))
U41_ggaa(T114, T115, T116, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, splitc51_in_ggaa(T115, T116))
U43_ggaa(T129, T130, T131, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, splitc51_in_ggaa(T130, T131))
U57_ga(T163, T164, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, T168, qsortc79_in_ga(T167))
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U42_ggaa(T114, T115, T116, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U44_ggaa(T129, T130, T131, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U58_ga(T163, T164, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, T171, qsortc79_in_ga(T168))
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
U59_ga(T163, T164, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, appendc94_in_ggga(T171, T163, T172))
U60_ga(T163, T164, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
appendc94_in_ggga([], T185, T186) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198) → U61_ggga(T195, T196, T197, T198, appendc94_in_ggga(T196, T197, T198))
U61_ggga(T195, T196, T197, T198, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))

The set Q consists of the following terms:

splitc51_in_ggaa(x0, x1)
qsortc79_in_ga(x0)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U57_ga(x0, x1, x2)
lec36_in_gg(x0, x1)
U42_ggaa(x0, x1, x2, x3)
gtc67_in_gg(x0, x1)
U44_ggaa(x0, x1, x2, x3)
U58_ga(x0, x1, x2, x3)
U40_gg(x0, x1, x2)
U45_gg(x0, x1, x2)
U59_ga(x0, x1, x2, x3)
U60_ga(x0, x1, x2)
appendc94_in_ggga(x0, x1, x2)
U61_ggga(x0, x1, x2, x3, x4)

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

(44) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(45) TRUE

(46) 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:

qsortc14_in_a([]) → qsortc14_out_a([])
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
qsortc1_in_ga([], []) → qsortc1_out_ga([], [])
qsortc1_in_ga(.(T14, []), T9) → U46_ga(T14, T9, qsortc14_in_a(T17))
U46_ga(T14, T9, qsortc14_out_a(T17)) → U47_ga(T14, T9, T17, qsortc14_in_a(T20))
U47_ga(T14, T9, T17, qsortc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, .(T33, T34)) → appendc23_out_ggga([], T33, T34, .(T33, T34))
appendc23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsortc1_out_ga(.(T14, []), T9)
qsortc1_in_ga(.(T69, .(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U50_ga(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U51_ga(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
qsortc1_in_ga(.(T215, .(T213, T214)), T9) → U53_ga(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U53_ga(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U54_ga(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U54_ga(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U55_ga(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U55_ga(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U56_ga(T215, T213, T214, T9, qc78_in_gagga(.(T213, T223), X13, T226, T215, T9))
qc78_in_gagga(T94, T158, T155, T69, T9) → U62_gagga(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U62_gagga(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U63_gagga(T94, T158, T155, T69, T9, appendc23_in_ggga(T155, T69, T158, T9))
U63_gagga(T94, T158, T155, T69, T9, appendc23_out_ggga(T155, T69, T158, T9)) → qc78_out_gagga(T94, T158, T155, T69, T9)
U56_ga(T215, T213, T214, T9, qc78_out_gagga(.(T213, T223), X13, T226, T215, T9)) → qsortc1_out_ga(.(T215, .(T213, T214)), T9)
U51_ga(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T94, X13, T155, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T94, X13, T155, T69, T9)) → qsortc1_out_ga(.(T69, .(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
qsortc14_in_a(x1)  =  qsortc14_in_a
qsortc14_out_a(x1)  =  qsortc14_out_a(x1)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsortc1_in_ga(x1, x2)  =  qsortc1_in_ga(x1)
qsortc1_out_ga(x1, x2)  =  qsortc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_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

(47) UsableRulesProof (EQUIVALENT transformation)

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

(48) 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

(49) PiDPToQDPProof (SOUND transformation)

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

(50) 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.

(51) 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

(52) YES

(53) Obligation:

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

QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U25_GA(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U25_GA(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U27_GA(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_GA(.(T67, T93), X12)

The TRS R consists of the following rules:

qsortc14_in_a([]) → qsortc14_out_a([])
lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
qsortc79_in_ga([], []) → qsortc79_out_ga([], [])
qsortc79_in_ga(.(T163, T164), X265) → U57_ga(T163, T164, X265, splitc51_in_ggaa(T164, T163, T167, T168))
U57_ga(T163, T164, X265, splitc51_out_ggaa(T164, T163, T167, T168)) → U58_ga(T163, T164, X265, T168, qsortc79_in_ga(T167, T171))
U58_ga(T163, T164, X265, T168, qsortc79_out_ga(T167, T171)) → U59_ga(T163, T164, X265, T171, qsortc79_in_ga(T168, T172))
U59_ga(T163, T164, X265, T171, qsortc79_out_ga(T168, T172)) → U60_ga(T163, T164, X265, appendc94_in_ggga(T171, T163, T172, X265))
appendc94_in_ggga([], T185, T186, .(T185, T186)) → appendc94_out_ggga([], T185, T186, .(T185, T186))
appendc94_in_ggga(.(T195, T196), T197, T198, .(T195, X295)) → U61_ggga(T195, T196, T197, T198, X295, appendc94_in_ggga(T196, T197, T198, X295))
U61_ggga(T195, T196, T197, T198, X295, appendc94_out_ggga(T196, T197, T198, X295)) → appendc94_out_ggga(.(T195, T196), T197, T198, .(T195, X295))
U60_ga(T163, T164, X265, appendc94_out_ggga(T171, T163, T172, X265)) → qsortc79_out_ga(.(T163, T164), X265)
qsortc1_in_ga([], []) → qsortc1_out_ga([], [])
qsortc1_in_ga(.(T14, []), T9) → U46_ga(T14, T9, qsortc14_in_a(T17))
U46_ga(T14, T9, qsortc14_out_a(T17)) → U47_ga(T14, T9, T17, qsortc14_in_a(T20))
U47_ga(T14, T9, T17, qsortc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, .(T33, T34)) → appendc23_out_ggga([], T33, T34, .(T33, T34))
appendc23_in_ggga(.(T45, T46), T47, T48, .(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(.(T45, T46), T47, T48, .(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsortc1_out_ga(.(T14, []), T9)
qsortc1_in_ga(.(T69, .(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U50_ga(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → U51_ga(T69, T67, T68, T9, T94, qsortc1_in_ga(.(T67, T93), T155))
qsortc1_in_ga(.(T215, .(T213, T214)), T9) → U53_ga(T215, T213, T214, T9, gtc67_in_gg(T213, T215))
U53_ga(T215, T213, T214, T9, gtc67_out_gg(T213, T215)) → U54_ga(T215, T213, T214, T9, splitc51_in_ggaa(T214, T215, T222, T223))
U54_ga(T215, T213, T214, T9, splitc51_out_ggaa(T214, T215, T222, T223)) → U55_ga(T215, T213, T214, T9, T223, qsortc79_in_ga(T222, T226))
U55_ga(T215, T213, T214, T9, T223, qsortc79_out_ga(T222, T226)) → U56_ga(T215, T213, T214, T9, qc78_in_gagga(.(T213, T223), X13, T226, T215, T9))
qc78_in_gagga(T94, T158, T155, T69, T9) → U62_gagga(T94, T158, T155, T69, T9, qsortc79_in_ga(T94, T158))
U62_gagga(T94, T158, T155, T69, T9, qsortc79_out_ga(T94, T158)) → U63_gagga(T94, T158, T155, T69, T9, appendc23_in_ggga(T155, T69, T158, T9))
U63_gagga(T94, T158, T155, T69, T9, appendc23_out_ggga(T155, T69, T158, T9)) → qc78_out_gagga(T94, T158, T155, T69, T9)
U56_ga(T215, T213, T214, T9, qc78_out_gagga(.(T213, T223), X13, T226, T215, T9)) → qsortc1_out_ga(.(T215, .(T213, T214)), T9)
U51_ga(T69, T67, T68, T9, T94, qsortc1_out_ga(.(T67, T93), T155)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T94, X13, T155, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T94, X13, T155, T69, T9)) → qsortc1_out_ga(.(T69, .(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
qsortc14_in_a(x1)  =  qsortc14_in_a
qsortc14_out_a(x1)  =  qsortc14_out_a(x1)
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsortc79_in_ga(x1, x2)  =  qsortc79_in_ga(x1)
qsortc79_out_ga(x1, x2)  =  qsortc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsortc1_in_ga(x1, x2)  =  qsortc1_in_ga(x1)
qsortc1_out_ga(x1, x2)  =  qsortc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)

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

(54) UsableRulesProof (EQUIVALENT transformation)

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

(55) Obligation:

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

QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U25_GA(T69, T67, T68, T9, lec36_in_gg(T67, T69))
U25_GA(T69, T67, T68, T9, lec36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, T9, splitc51_in_ggaa(T68, T69, T93, T94))
U27_GA(T69, T67, T68, T9, splitc51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_GA(.(T67, T93), X12)

The TRS R consists of the following rules:

lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
splitc51_in_ggaa([], T101, [], []) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U41_ggaa(T114, T115, T116, X167, X168, lec36_in_gg(T114, T116))
splitc51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U43_ggaa(T129, T130, T131, X199, X200, gtc67_in_gg(T129, T131))
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
U41_ggaa(T114, T115, T116, X167, X168, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, X167, X168, splitc51_in_ggaa(T115, T116, X167, X168))
U43_ggaa(T129, T130, T131, X199, X200, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, X199, X200, splitc51_in_ggaa(T130, T131, X199, X200))
U42_ggaa(T114, T115, T116, X167, X168, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U44_ggaa(T129, T130, T131, X199, X200, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
s(x1)  =  s(x1)
lec36_in_gg(x1, x2)  =  lec36_in_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
0  =  0
lec36_out_gg(x1, x2)  =  lec36_out_gg(x1, x2)
gtc67_in_gg(x1, x2)  =  gtc67_in_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
gtc67_out_gg(x1, x2)  =  gtc67_out_gg(x1, x2)
splitc51_in_ggaa(x1, x2, x3, x4)  =  splitc51_in_ggaa(x1, x2)
splitc51_out_ggaa(x1, x2, x3, x4)  =  splitc51_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)

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

(56) PiDPToQDPProof (SOUND transformation)

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

(57) Obligation:

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

QSORT1_IN_GA(.(T69, .(T67, T68))) → U25_GA(T69, T67, T68, lec36_in_gg(T67, T69))
U25_GA(T69, T67, T68, lec36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, splitc51_in_ggaa(T68, T69))
U27_GA(T69, T67, T68, splitc51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_GA(.(T67, T93))

The TRS R consists of the following rules:

lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
splitc51_in_ggaa([], T101) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116) → U41_ggaa(T114, T115, T116, lec36_in_gg(T114, T116))
splitc51_in_ggaa(.(T129, T130), T131) → U43_ggaa(T129, T130, T131, gtc67_in_gg(T129, T131))
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
U41_ggaa(T114, T115, T116, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, splitc51_in_ggaa(T115, T116))
U43_ggaa(T129, T130, T131, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, splitc51_in_ggaa(T130, T131))
U42_ggaa(T114, T115, T116, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U44_ggaa(T129, T130, T131, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))

The set Q consists of the following terms:

lec36_in_gg(x0, x1)
splitc51_in_ggaa(x0, x1)
U40_gg(x0, x1, x2)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U42_ggaa(x0, x1, x2, x3)
gtc67_in_gg(x0, x1)
U44_ggaa(x0, x1, x2, x3)
U45_gg(x0, x1, x2)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QSORT1_IN_GA(.(T69, .(T67, T68))) → U25_GA(T69, T67, T68, lec36_in_gg(T67, T69))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(0) = 0   
POL(QSORT1_IN_GA(x₁)) = x₁   
POL(U25_GA(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U27_GA(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U40_gg(x₁, x₂, x₃)) = 0   
POL(U41_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U42_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U43_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U44_ggaa(x₁, x₂, x₃, x₄)) = x₄   
POL(U45_gg(x₁, x₂, x₃)) = 0   
POL([]) = 0   
POL(gtc67_in_gg(x₁, x₂)) = 0   
POL(gtc67_out_gg(x₁, x₂)) = 0   
POL(lec36_in_gg(x₁, x₂)) = 0   
POL(lec36_out_gg(x₁, x₂)) = 0   
POL(s(x₁)) = 0   
POL(splitc51_in_ggaa(x₁, x₂)) = x₁   
POL(splitc51_out_ggaa(x₁, x₂, x₃, x₄)) = x₃   

The following usable rules [FROCOS05] were oriented:

splitc51_in_ggaa([], T101) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116) → U41_ggaa(T114, T115, T116, lec36_in_gg(T114, T116))
splitc51_in_ggaa(.(T129, T130), T131) → U43_ggaa(T129, T130, T131, gtc67_in_gg(T129, T131))
U41_ggaa(T114, T115, T116, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, splitc51_in_ggaa(T115, T116))
U42_ggaa(T114, T115, T116, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U43_ggaa(T129, T130, T131, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, splitc51_in_ggaa(T130, T131))
U44_ggaa(T129, T130, T131, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))

(59) Obligation:

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

U25_GA(T69, T67, T68, lec36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, splitc51_in_ggaa(T68, T69))
U27_GA(T69, T67, T68, splitc51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_GA(.(T67, T93))

The TRS R consists of the following rules:

lec36_in_gg(s(T82), s(T83)) → U40_gg(T82, T83, lec36_in_gg(T82, T83))
lec36_in_gg(0, s(T90)) → lec36_out_gg(0, s(T90))
lec36_in_gg(0, 0) → lec36_out_gg(0, 0)
splitc51_in_ggaa([], T101) → splitc51_out_ggaa([], T101, [], [])
splitc51_in_ggaa(.(T114, T115), T116) → U41_ggaa(T114, T115, T116, lec36_in_gg(T114, T116))
splitc51_in_ggaa(.(T129, T130), T131) → U43_ggaa(T129, T130, T131, gtc67_in_gg(T129, T131))
U40_gg(T82, T83, lec36_out_gg(T82, T83)) → lec36_out_gg(s(T82), s(T83))
U41_ggaa(T114, T115, T116, lec36_out_gg(T114, T116)) → U42_ggaa(T114, T115, T116, splitc51_in_ggaa(T115, T116))
U43_ggaa(T129, T130, T131, gtc67_out_gg(T129, T131)) → U44_ggaa(T129, T130, T131, splitc51_in_ggaa(T130, T131))
U42_ggaa(T114, T115, T116, splitc51_out_ggaa(T115, T116, X167, X168)) → splitc51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
gtc67_in_gg(s(T144), s(T145)) → U45_gg(T144, T145, gtc67_in_gg(T144, T145))
gtc67_in_gg(s(T150), 0) → gtc67_out_gg(s(T150), 0)
U44_ggaa(T129, T130, T131, splitc51_out_ggaa(T130, T131, X199, X200)) → splitc51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U45_gg(T144, T145, gtc67_out_gg(T144, T145)) → gtc67_out_gg(s(T144), s(T145))

The set Q consists of the following terms:

lec36_in_gg(x0, x1)
splitc51_in_ggaa(x0, x1)
U40_gg(x0, x1, x2)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U42_ggaa(x0, x1, x2, x3)
gtc67_in_gg(x0, x1)
U44_ggaa(x0, x1, x2, x3)
U45_gg(x0, x1, x2)

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

(60) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(61) TRUE