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

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 284 ms)

↳2 TRIPLES

↳3 UndefinedPredicateInTriplesTransformerProof (⇒, 0 ms)

↳4 TRIPLES

↳5 TriplesToPiDPProof (⇒, 482 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 15 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇒, 30 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇔, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔, 0 ms)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇔, 0 ms)

        ↳27 QDP

        ↳28 QDPSizeChangeProof (⇔, 0 ms)

        ↳29 YES

    ↳30 PiDP

        ↳31 UsableRulesProof (⇔, 0 ms)

        ↳32 PiDP

        ↳33 PiDPToQDPProof (⇒, 0 ms)

        ↳34 QDP

        ↳35 QDPSizeChangeProof (⇔, 0 ms)

        ↳36 YES

    ↳37 PiDP

        ↳38 UsableRulesProof (⇔, 0 ms)

        ↳39 PiDP

        ↳40 PiDPToQDPProof (⇒, 0 ms)

        ↳41 QDP

        ↳42 QDPOrderProof (⇔, 232 ms)

        ↳43 QDP

        ↳44 DependencyGraphProof (⇔, 0 ms)

        ↳45 TRUE

    ↳46 PiDP

        ↳47 UsableRulesProof (⇔, 0 ms)

        ↳48 PiDP

        ↳49 PiDPToQDPProof (⇒, 0 ms)

        ↳50 QDP

        ↳51 QDPSizeChangeProof (⇔, 0 ms)

        ↳52 YES

    ↳53 PiDP

        ↳54 UsableRulesProof (⇔, 0 ms)

        ↳55 PiDP

        ↳56 PiDPToQDPProof (⇒, 0 ms)

        ↳57 QDP

        ↳58 QDPOrderProof (⇔, 146 ms)

        ↳59 QDP

        ↳60 DependencyGraphProof (⇔, 0 ms)

        ↳61 TRUE

(0) Obligation:

Clauses:

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

Query: qsort(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="F: 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="A: qsort([], X12)\n\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(qsort([], X13), append(T19, .(T14, X13), T9))\n\nT14 is ground\nT19 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="B: append(T19, .(T14, T24), T9)\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: append(T19, .(T14, T24), T9), SCOPE: 4, CLAUSE: 5 | append(T19, .(T14, T24), T9), SCOPE: 4, CLAUSE: 6\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: append(T19, .(T14, T24), T9), SCOPE: 4, CLAUSE: 5\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: append(T19, .(T14, T24), T9), SCOPE: 4, CLAUSE: 6\n\nT14 is ground\nT19 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
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(T50, .(T51, T52), T54)\n\nT50 is ground\nT51 is ground\nT52 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(T71, T73), split(T72, T73, X84, X85)), ','(qsort(.(T71, X84), X12), ','(qsort(X85, X13), append(X12, .(T73, X13), T9))))\n\nT71 is ground\nT72 is ground\nT73 is ground\nX12 is free\nX13 is free\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="C: le(T71, T73)\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(split(T72, T73, X84, X85), ','(qsort(.(T71, X84), X12), ','(qsort(X85, X13), append(X12, .(T73, X13), T9))))\n\nT71 is ground\nT72 is ground\nT73 is ground\nX12 is free\nX13 is free\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: le(T71, T73), SCOPE: 5, CLAUSE: 9 | le(T71, T73), SCOPE: 5, CLAUSE: 10 | le(T71, T73), SCOPE: 5, CLAUSE: 11\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: le(T71, T73), SCOPE: 5, CLAUSE: 9\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: le(T71, T73), SCOPE: 5, CLAUSE: 10 | le(T71, T73), SCOPE: 5, CLAUSE: 11\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: le(T86, T87)\n\nT86 is ground\nT87 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: le(T71, T73), SCOPE: 5, CLAUSE: 10\n\nT71 is ground\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: le(T71, T73), SCOPE: 5, CLAUSE: 11\n\nT71 is ground\nT73 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="D: split(T72, T73, X84, X85)\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(qsort(.(T71, T97), X12), ','(qsort(T98, X13), append(X12, .(T73, X13), T9)))\n\nT71 is ground\nT73 is ground\nT97 is ground\nT98 is ground\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 2 | split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 3 | split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 4\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 2\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 3 | split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 4\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
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(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 3\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: split(T72, T73, X84, X85), SCOPE: 6, CLAUSE: 4\n\nT72 is ground\nT73 is ground\nX84 is free\nX85 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(le(T118, T120), split(T119, T120, X149, X150))\n\nT118 is ground\nT119 is ground\nT120 is ground\nX149 is free\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: le(T118, T120)\n\nT118 is ground\nT120 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: split(T119, T120, X149, X150)\n\nT119 is ground\nT120 is ground\nX149 is free\nX150 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(gt(T133, T135), split(T134, T135, X176, X177))\n\nT133 is ground\nT134 is ground\nT135 is ground\nX176 is free\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="E: gt(T133, T135)\n\nT133 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: split(T134, T135, X176, X177)\n\nT134 is ground\nT135 is ground\nX176 is free\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: gt(T133, T135), SCOPE: 7, CLAUSE: 7 | gt(T133, T135), SCOPE: 7, CLAUSE: 8\n\nT133 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: gt(T133, T135), SCOPE: 7, CLAUSE: 7\n\nT133 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: gt(T133, T135), SCOPE: 7, CLAUSE: 8\n\nT133 is ground\nT135 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: gt(T148, T149)\n\nT148 is ground\nT149 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(.(T71, T97), X12)\n\nT71 is ground\nT97 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="G: ','(qsort(T98, X13), append(T157, .(T73, X13), T9))\n\nT73 is ground\nT98 is ground\nT157 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="H: qsort(T98, X13)\n\nT98 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: append(T157, .(T73, T158), T9)\n\nT73 is ground\nT157 is ground\nT158 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: qsort(T98, X13), SCOPE: 8, CLAUSE: 0 | qsort(T98, X13), SCOPE: 8, CLAUSE: 1\n\nT98 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: qsort(T98, X13), SCOPE: 8, CLAUSE: 0\n\nT98 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: qsort(T98, X13), SCOPE: 8, CLAUSE: 1\n\nT98 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, X214, X215), ','(qsort(X214, X216), ','(qsort(X215, X217), append(X216, .(T163, X217), X218))))\n\nT163 is ground\nT164 is ground\nX218 is free\nX214 is free\nX215 is free\nX216 is free\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: split(T164, T163, X214, X215)\n\nT163 is ground\nT164 is ground\nX214 is free\nX215 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(qsort(T167, X216), ','(qsort(T168, X217), append(X216, .(T163, X217), X218)))\n\nT163 is ground\nT167 is ground\nT168 is ground\nX218 is free\nX216 is free\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: qsort(T167, X216)\n\nT167 is ground\nX216 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(qsort(T168, X217), append(T171, .(T163, X217), X218))\n\nT163 is ground\nT168 is ground\nT171 is ground\nX218 is free\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: qsort(T168, X217)\n\nT168 is ground\nX217 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="I: append(T171, .(T163, T172), X218)\n\nT163 is ground\nT171 is ground\nT172 is ground\nX218 is free", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: append(T171, .(T163, T172), X218), SCOPE: 9, CLAUSE: 5 | append(T171, .(T163, T172), X218), SCOPE: 9, CLAUSE: 6\n\nT163 is ground\nT171 is ground\nT172 is ground\nX218 is free", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: append(T171, .(T163, T172), X218), SCOPE: 9, CLAUSE: 5\n\nT163 is ground\nT171 is ground\nT172 is ground\nX218 is free", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: append(T171, .(T163, T172), X218), SCOPE: 9, CLAUSE: 6\n\nT163 is ground\nT171 is ground\nT172 is ground\nX218 is free", fontsize=16, style=filled, fillcolor=lightyellow];
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), X244)\n\nT196 is ground\nT197 is ground\nT198 is ground\nX244 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, X268, X269)), ','(qsort(X268, X12), ','(qsort(.(T213, X269), X13), append(X12, .(T215, X13), T9))))\n\nT213 is ground\nT214 is ground\nT215 is ground\nX12 is free\nX13 is free\nX268 is free\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
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, X268, X269), ','(qsort(X268, X12), ','(qsort(.(T213, X269), X13), append(X12, .(T215, X13), T9))))\n\nT213 is ground\nT214 is ground\nT215 is ground\nX12 is free\nX13 is free\nX268 is free\nX269 is free", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: split(T214, T215, X268, X269)\n\nT214 is ground\nT215 is ground\nX268 is free\nX269 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 substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 112 [arrowhead = none ];
112 -> 3;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionX7 -> 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 = lightgreen];
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 = lightgreen];
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([], X18, [], []).\nand substitutionT7 -> [],\nT6 -> T14,\nX18 -> T14,\nX10 -> [],\nX11 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 121 [arrowhead = none ];
121 -> 12;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT19 is ground\nreplacements:X12 -> T19", 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:\nT24 is ground\nreplacements:X13 -> T24", 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 substitutionX12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
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([], X36, X36).\nand substitutionT19 -> [],\nT14 -> T37,\nT24 -> T38,\nX36 -> .(T37, T38),\nT9 -> .(T37, T38)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 139 [arrowhead = none ];
139 -> 27;

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

141 [label="EVAL with clause\nappend(.(X45, X46), X47, .(X45, X48)) :- append(X46, X47, X48).\nand substitutionX45 -> T49,\nX46 -> T50,\nT19 -> .(T49, T50),\nT14 -> T51,\nT24 -> T52,\nX47 -> .(T51, T52),\nX48 -> T54,\nT9 -> .(T49, T54),\nT53 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 141 [arrowhead = none ];
141 -> 30;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT19 -> T50\nT14 -> T51\nT24 -> T52\nT9 -> T54", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 144 [arrowhead = none , style=dashed];
144 -> 23[style=dashed];

145 [label="EVAL with clause\nsplit(.(X79, X80), X81, .(X79, X82), X83) :- ','(le(X79, X81), split(X80, X81, X82, X83)).\nand substitutionX79 -> T71,\nX80 -> T72,\nT7 -> .(T71, T72),\nT6 -> T73,\nX81 -> T73,\nX82 -> X84,\nX10 -> .(T71, X84),\nX11 -> X85,\nX83 -> X85", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 145 [arrowhead = none ];
145 -> 34;

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

147 [label="EVAL with clause\nsplit(.(X263, X264), X265, X266, .(X263, X267)) :- ','(gt(X263, X265), split(X264, X265, X266, X267)).\nand substitutionX263 -> T213,\nX264 -> T214,\nT7 -> .(T213, T214),\nT6 -> T215,\nX265 -> T215,\nX10 -> X268,\nX266 -> X268,\nX267 -> X269,\nX11 -> .(T213, X269)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 147 [arrowhead = none ];
147 -> 103;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT71 is ground\nT73 is ground", 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:\nT72 is ground\nT73 is ground\nT97 is ground\nT98 is ground\nreplacements:X84 -> T97,\nX85 -> T98", 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(X98), s(X99)) :- le(X98, X99).\nand substitutionX98 -> T86,\nT71 -> s(T86),\nX99 -> T87,\nT73 -> s(T87)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 156 [arrowhead = none ];
156 -> 41;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT71 -> T86\nT73 -> T87", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 160 [arrowhead = none , style=dashed];
160 -> 36[style=dashed];

161 [label="EVAL with clause\nle(0, s(X106)).\nand substitutionT71 -> 0,\nX106 -> T94,\nT73 -> s(T94)", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 161 [arrowhead = none ];
161 -> 45;

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

163 [label="EVAL with clause\nle(0, 0).\nand substitutionT71 -> 0,\nT73 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 163 [arrowhead = none ];
163 -> 48;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT71 is ground\nT97 is ground\nT157 is ground\nreplacements:X12 -> T157", 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([], X115, [], []).\nand substitutionT72 -> [],\nT73 -> T105,\nX115 -> T105,\nX84 -> [],\nX85 -> []", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 172 [arrowhead = none ];
172 -> 56;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(X144, X145), X146, .(X144, X147), X148) :- ','(le(X144, X146), split(X145, X146, X147, X148)).\nand substitutionX144 -> T118,\nX145 -> T119,\nT72 -> .(T118, T119),\nT73 -> T120,\nX146 -> T120,\nX147 -> X149,\nX84 -> .(T118, X149),\nX85 -> X150,\nX148 -> X150", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 177 [arrowhead = none ];
177 -> 61;

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

179 [label="EVAL with clause\nsplit(.(X171, X172), X173, X174, .(X171, X175)) :- ','(gt(X171, X173), split(X172, X173, X174, X175)).\nand substitutionX171 -> T133,\nX172 -> T134,\nT72 -> .(T133, T134),\nT73 -> T135,\nX173 -> T135,\nX84 -> X176,\nX174 -> X176,\nX175 -> X177,\nX85 -> .(T133, X177)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 179 [arrowhead = none ];
179 -> 65;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT118 is ground\nT120 is ground", fontsize=14, style = filled, fillcolor = violet];
61 -> 182 [arrowhead = none ];
182 -> 64;

183 [label="INSTANCE with matching:\nT71 -> T118\nT73 -> T120", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 183 [arrowhead = none , style=dashed];
183 -> 36[style=dashed];

184 [label="INSTANCE with matching:\nT72 -> T119\nT73 -> T120\nX84 -> X149\nX85 -> X150", 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\nnew knowledge:\nT133 is ground\nT135 is ground", 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:\nT72 -> T134\nT73 -> T135\nX84 -> X176\nX85 -> X177", 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(X190), s(X191)) :- gt(X190, X191).\nand substitutionX190 -> T148,\nT133 -> s(T148),\nX191 -> T149,\nT135 -> s(T149)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 191 [arrowhead = none ];
191 -> 72;

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

193 [label="EVAL with clause\ngt(s(X196), 0).\nand substitutionX196 -> T154,\nT133 -> s(T154),\nT135 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 193 [arrowhead = none ];
193 -> 74;

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

195 [label="INSTANCE with matching:\nT133 -> T148\nT135 -> T149", 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 -> .(T71, T97)\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:\nT98 is ground\nT158 is ground\nreplacements:X13 -> 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:\nT19 -> T157\nT14 -> T73\nT24 -> 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 substitutionT98 -> [],\nX13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 204 [arrowhead = none ];
204 -> 84;

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

206 [label="EVAL with clause\nqsort(.(X211, X212), X213) :- ','(split(X212, X211, X214, X215), ','(qsort(X214, X216), ','(qsort(X215, X217), append(X216, .(X211, X217), X213)))).\nand substitutionX211 -> T163,\nX212 -> T164,\nT98 -> .(T163, T164),\nX13 -> X218,\nX213 -> X218", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 206 [arrowhead = none ];
206 -> 87;

207 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT164 is ground\nT163 is ground\nT167 is ground\nT168 is ground\nreplacements:X214 -> T167,\nX215 -> T168", fontsize=14, style = filled, fillcolor = violet];
87 -> 210 [arrowhead = none ];
210 -> 90;

211 [label="INSTANCE with matching:\nT72 -> T164\nT73 -> T163\nX84 -> X214\nX85 -> X215", 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:\nT167 is ground\nT171 is ground\nreplacements:X216 -> T171", fontsize=14, style = filled, fillcolor = violet];
90 -> 213 [arrowhead = none ];
213 -> 92;

214 [label="INSTANCE with matching:\nT98 -> T167\nX13 -> X216", 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:\nT168 is ground\nT172 is ground\nreplacements:X217 -> T172", fontsize=14, style = filled, fillcolor = violet];
92 -> 216 [arrowhead = none ];
216 -> 94;

217 [label="INSTANCE with matching:\nT98 -> T168\nX13 -> X217", 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([], X229, X229).\nand substitutionT171 -> [],\nT163 -> T185,\nT172 -> T186,\nX229 -> .(T185, T186),\nX218 -> .(T185, T186)", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 221 [arrowhead = none ];
221 -> 98;

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

223 [label="EVAL with clause\nappend(.(X240, X241), X242, .(X240, X243)) :- append(X241, X242, X243).\nand substitutionX240 -> T195,\nX241 -> T196,\nT171 -> .(T195, T196),\nT163 -> T197,\nT172 -> T198,\nX242 -> .(T197, T198),\nX243 -> X244,\nX218 -> .(T195, X244)", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 223 [arrowhead = none ];
223 -> 101;

224 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nX218 -> X244", 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\nnew knowledge:\nT213 is ground\nT215 is ground", fontsize=14, style = filled, fillcolor = violet];
103 -> 228 [arrowhead = none ];
228 -> 106;

229 [label="INSTANCE with matching:\nT133 -> T213\nT135 -> 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:\nT214 is ground\nT215 is ground\nT222 is ground\nT223 is ground\nreplacements:X268 -> T222,\nX269 -> T223", fontsize=14, style = filled, fillcolor = violet];
106 -> 231 [arrowhead = none ];
231 -> 108;

232 [label="INSTANCE with matching:\nT72 -> T214\nT73 -> T215\nX84 -> X268\nX85 -> X269", 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:\nT222 is ground\nT226 is ground\nreplacements:X12 -> T226", fontsize=14, style = filled, fillcolor = violet];
108 -> 234 [arrowhead = none ];
234 -> 110;

235 [label="INSTANCE with matching:\nT98 -> T222\nX13 -> X12", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 235 [arrowhead = none , style=dashed];
235 -> 79[style=dashed];

236 [label="INSTANCE with matching:\nT98 -> .(T213, T223)\nT157 -> T226\nT73 -> T215", fontsize=14, style = filled, fillcolor = lightgrey];
110 -> 236 [arrowhead = none , style=dashed];
236 -> 78[style=dashed];

}

(2) Obligation:

Triples:

appendB(.(X1, X2), X3, X4, .(X1, X5)) :- appendB(X2, X3, X4, X5).
leC(s(X1), s(X2)) :- leC(X1, X2).
splitD(.(X1, X2), X3, .(X1, X4), X5) :- leC(X1, X3).
splitD(.(X1, X2), X3, .(X1, X4), X5) :- ','(lecC(X1, X3), splitD(X2, X3, X4, X5)).
splitD(.(X1, X2), X3, X4, .(X1, X5)) :- gtE(X1, X3).
splitD(.(X1, X2), X3, X4, .(X1, X5)) :- ','(gtcE(X1, X3), splitD(X2, X3, X4, X5)).
gtE(s(X1), s(X2)) :- gtE(X1, X2).
qsortH(.(X1, X2), X3) :- splitD(X2, X1, X4, X5).
qsortH(.(X1, X2), X3) :- ','(splitcD(X2, X1, X4, X5), qsortH(X4, X6)).
qsortH(.(X1, X2), X3) :- ','(splitcD(X2, X1, X4, X5), ','(qsortcH(X4, X6), qsortH(X5, X7))).
qsortH(.(X1, X2), X3) :- ','(splitcD(X2, X1, X4, X5), ','(qsortcH(X4, X6), ','(qsortcH(X5, X7), appendI(X6, X1, X7, X3)))).
appendI(.(X1, X2), X3, X4, .(X1, X5)) :- appendI(X2, X3, X4, X5).
pG(X1, X2, X3, X4, X5) :- qsortH(X1, X2).
pG(X1, X2, X3, X4, X5) :- ','(qsortcH(X1, X2), appendB(X3, X4, X2, X5)).
qsortF(.(X1, []), X2) :- qsortA(X3).
qsortF(.(X1, []), X2) :- ','(qsortcA(X3), qsortA(X4)).
qsortF(.(X1, []), X2) :- ','(qsortcA(X3), ','(qsortcA(X4), appendB(X3, X1, X4, X2))).
qsortF(.(X1, .(X2, X3)), X4) :- leC(X2, X1).
qsortF(.(X1, .(X2, X3)), X4) :- ','(lecC(X2, X1), splitD(X3, X1, X5, X6)).
qsortF(.(X1, .(X2, X3)), X4) :- ','(lecC(X2, X1), ','(splitcD(X3, X1, X5, X6), qsortF(.(X2, X5), X7))).
qsortF(.(X1, .(X2, X3)), X4) :- ','(lecC(X2, X1), ','(splitcD(X3, X1, X5, X6), ','(qsortcF(.(X2, X5), X7), pG(X6, X8, X7, X1, X4)))).
qsortF(.(X1, .(X2, X3)), X4) :- gtE(X2, X1).
qsortF(.(X1, .(X2, X3)), X4) :- ','(gtcE(X2, X1), splitD(X3, X1, X5, X6)).
qsortF(.(X1, .(X2, X3)), X4) :- ','(gtcE(X2, X1), ','(splitcD(X3, X1, X5, X6), qsortH(X5, X7))).
qsortF(.(X1, .(X2, X3)), X4) :- ','(gtcE(X2, X1), ','(splitcD(X3, X1, X5, X6), ','(qsortcH(X5, X7), pG(.(X2, X6), X8, X7, X1, X4)))).

Clauses:

qsortcA([]).
appendcB([], X1, X2, .(X1, X2)).
appendcB(.(X1, X2), X3, X4, .(X1, X5)) :- appendcB(X2, X3, X4, X5).
lecC(s(X1), s(X2)) :- lecC(X1, X2).
lecC(0, s(X1)).
lecC(0, 0).
splitcD([], X1, [], []).
splitcD(.(X1, X2), X3, .(X1, X4), X5) :- ','(lecC(X1, X3), splitcD(X2, X3, X4, X5)).
splitcD(.(X1, X2), X3, X4, .(X1, X5)) :- ','(gtcE(X1, X3), splitcD(X2, X3, X4, X5)).
gtcE(s(X1), s(X2)) :- gtcE(X1, X2).
gtcE(s(X1), 0).
qsortcF([], []).
qsortcF(.(X1, []), X2) :- ','(qsortcA(X3), ','(qsortcA(X4), appendcB(X3, X1, X4, X2))).
qsortcF(.(X1, .(X2, X3)), X4) :- ','(lecC(X2, X1), ','(splitcD(X3, X1, X5, X6), ','(qsortcF(.(X2, X5), X7), qcG(X6, X8, X7, X1, X4)))).
qsortcF(.(X1, .(X2, X3)), X4) :- ','(gtcE(X2, X1), ','(splitcD(X3, X1, X5, X6), ','(qsortcH(X5, X7), qcG(.(X2, X6), X8, X7, X1, X4)))).
qsortcH([], []).
qsortcH(.(X1, X2), X3) :- ','(splitcD(X2, X1, X4, X5), ','(qsortcH(X4, X6), ','(qsortcH(X5, X7), appendcI(X6, X1, X7, X3)))).
appendcI([], X1, X2, .(X1, X2)).
appendcI(.(X1, X2), X3, X4, .(X1, X5)) :- appendcI(X2, X3, X4, X5).
qcG(X1, X2, X3, X4, X5) :- ','(qsortcH(X1, X2), appendcB(X3, X4, X2, X5)).

Afs:

qsortF(x1, x2) = qsortF(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [DT09].

(4) Obligation:

Triples:

appendB(.(X1, X2), X3, X4, .(X1, X5)) :- appendB(X2, X3, X4, X5).
leC(s(X1), s(X2)) :- leC(X1, X2).
splitD(.(X1, X2), X3, .(X1, X4), X5) :- leC(X1, X3).
splitD(.(X1, X2), X3, .(X1, X4), X5) :- ','(lecC(X1, X3), splitD(X2, X3, X4, X5)).
splitD(.(X1, X2), X3, X4, .(X1, X5)) :- gtE(X1, X3).
splitD(.(X1, X2), X3, X4, .(X1, X5)) :- ','(gtcE(X1, X3), splitD(X2, X3, X4, X5)).
gtE(s(X1), s(X2)) :- gtE(X1, X2).
qsortH(.(X1, X2), X3) :- splitD(X2, X1, X4, X5).
qsortH(.(X1, X2), X3) :- ','(splitcD(X2, X1, X4, X5), qsortH(X4, X6)).
qsortH(.(X1, X2), X3) :- ','(splitcD(X2, X1, X4, X5), ','(qsortcH(X4, X6), qsortH(X5, X7))).
qsortH(.(X1, X2), X3) :- ','(splitcD(X2, X1, X4, X5), ','(qsortcH(X4, X6), ','(qsortcH(X5, X7), appendI(X6, X1, X7, X3)))).
appendI(.(X1, X2), X3, X4, .(X1, X5)) :- appendI(X2, X3, X4, X5).
pG(X1, X2, X3, X4, X5) :- qsortH(X1, X2).
pG(X1, X2, X3, X4, X5) :- ','(qsortcH(X1, X2), appendB(X3, X4, X2, X5)).
qsortF(.(X1, []), X2) :- ','(qsortcA(X3), ','(qsortcA(X4), appendB(X3, X1, X4, X2))).
qsortF(.(X1, .(X2, X3)), X4) :- leC(X2, X1).
qsortF(.(X1, .(X2, X3)), X4) :- ','(lecC(X2, X1), splitD(X3, X1, X5, X6)).
qsortF(.(X1, .(X2, X3)), X4) :- ','(lecC(X2, X1), ','(splitcD(X3, X1, X5, X6), qsortF(.(X2, X5), X7))).
qsortF(.(X1, .(X2, X3)), X4) :- ','(lecC(X2, X1), ','(splitcD(X3, X1, X5, X6), ','(qsortcF(.(X2, X5), X7), pG(X6, X8, X7, X1, X4)))).
qsortF(.(X1, .(X2, X3)), X4) :- gtE(X2, X1).
qsortF(.(X1, .(X2, X3)), X4) :- ','(gtcE(X2, X1), splitD(X3, X1, X5, X6)).
qsortF(.(X1, .(X2, X3)), X4) :- ','(gtcE(X2, X1), ','(splitcD(X3, X1, X5, X6), qsortH(X5, X7))).
qsortF(.(X1, .(X2, X3)), X4) :- ','(gtcE(X2, X1), ','(splitcD(X3, X1, X5, X6), ','(qsortcH(X5, X7), pG(.(X2, X6), X8, X7, X1, X4)))).

Clauses:

qsortcA([]).
appendcB([], X1, X2, .(X1, X2)).
appendcB(.(X1, X2), X3, X4, .(X1, X5)) :- appendcB(X2, X3, X4, X5).
lecC(s(X1), s(X2)) :- lecC(X1, X2).
lecC(0, s(X1)).
lecC(0, 0).
splitcD([], X1, [], []).
splitcD(.(X1, X2), X3, .(X1, X4), X5) :- ','(lecC(X1, X3), splitcD(X2, X3, X4, X5)).
splitcD(.(X1, X2), X3, X4, .(X1, X5)) :- ','(gtcE(X1, X3), splitcD(X2, X3, X4, X5)).
gtcE(s(X1), s(X2)) :- gtcE(X1, X2).
gtcE(s(X1), 0).
qsortcF([], []).
qsortcF(.(X1, []), X2) :- ','(qsortcA(X3), ','(qsortcA(X4), appendcB(X3, X1, X4, X2))).
qsortcF(.(X1, .(X2, X3)), X4) :- ','(lecC(X2, X1), ','(splitcD(X3, X1, X5, X6), ','(qsortcF(.(X2, X5), X7), qcG(X6, X8, X7, X1, X4)))).
qsortcF(.(X1, .(X2, X3)), X4) :- ','(gtcE(X2, X1), ','(splitcD(X3, X1, X5, X6), ','(qsortcH(X5, X7), qcG(.(X2, X6), X8, X7, X1, X4)))).
qsortcH([], []).
qsortcH(.(X1, X2), X3) :- ','(splitcD(X2, X1, X4, X5), ','(qsortcH(X4, X6), ','(qsortcH(X5, X7), appendcI(X6, X1, X7, X3)))).
appendcI([], X1, X2, .(X1, X2)).
appendcI(.(X1, X2), X3, X4, .(X1, X5)) :- appendcI(X2, X3, X4, X5).
qcG(X1, X2, X3, X4, X5) :- ','(qsortcH(X1, X2), appendcB(X3, X4, X2, X5)).

Afs:

qsortF(x1, x2) = qsortF(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
qsortF_in: (b,f)
appendB_in: (b,b,b,f)
leC_in: (b,b)
lecC_in: (b,b)
splitD_in: (b,b,f,f)
gtE_in: (b,b)
gtcE_in: (b,b)
splitcD_in: (b,b,f,f)
qsortH_in: (b,f)
qsortcH_in: (b,f)
appendcI_in: (b,b,b,f)
appendI_in: (b,b,b,f)
pG_in: (b,f,b,b,f)
qsortcF_in: (b,f)
appendcB_in: (b,b,b,f)
qcG_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:

QSORTF_IN_GA(.(X1, []), X2) → U21_GA(X1, X2, qsortcA_in_a(X3))
U21_GA(X1, X2, qsortcA_out_a(X3)) → U22_GA(X1, X2, X3, qsortcA_in_a(X4))
U22_GA(X1, X2, X3, qsortcA_out_a(X4)) → U23_GA(X1, X2, appendB_in_ggga(X3, X1, X4, X2))
U22_GA(X1, X2, X3, qsortcA_out_a(X4)) → APPENDB_IN_GGGA(X3, X1, X4, X2)
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U1_GGGA(X1, X2, X3, X4, X5, appendB_in_ggga(X2, X3, X4, X5))
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U24_GA(X1, X2, X3, X4, leC_in_gg(X2, X1))
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → LEC_IN_GG(X2, X1)
LEC_IN_GG(s(X1), s(X2)) → U2_GG(X1, X2, leC_in_gg(X1, X2))
LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U25_GA(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U25_GA(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U26_GA(X1, X2, X3, X4, splitD_in_ggaa(X3, X1, X5, X6))
U25_GA(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → SPLITD_IN_GGAA(X3, X1, X5, X6)
SPLITD_IN_GGAA(.(X1, X2), X3, .(X1, X4), X5) → U3_GGAA(X1, X2, X3, X4, X5, leC_in_gg(X1, X3))
SPLITD_IN_GGAA(.(X1, X2), X3, .(X1, X4), X5) → LEC_IN_GG(X1, X3)
SPLITD_IN_GGAA(.(X1, X2), X3, .(X1, X4), X5) → U4_GGAA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U4_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U5_GGAA(X1, X2, X3, X4, X5, splitD_in_ggaa(X2, X3, X4, X5))
U4_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3, X4, X5)
SPLITD_IN_GGAA(.(X1, X2), X3, X4, .(X1, X5)) → U6_GGAA(X1, X2, X3, X4, X5, gtE_in_gg(X1, X3))
SPLITD_IN_GGAA(.(X1, X2), X3, X4, .(X1, X5)) → GTE_IN_GG(X1, X3)
GTE_IN_GG(s(X1), s(X2)) → U9_GG(X1, X2, gtE_in_gg(X1, X2))
GTE_IN_GG(s(X1), s(X2)) → GTE_IN_GG(X1, X2)
SPLITD_IN_GGAA(.(X1, X2), X3, X4, .(X1, X5)) → U7_GGAA(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U7_GGAA(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U8_GGAA(X1, X2, X3, X4, X5, splitD_in_ggaa(X2, X3, X4, X5))
U7_GGAA(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3, X4, X5)
U25_GA(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U27_GA(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U27_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U28_GA(X1, X2, X3, X4, qsortF_in_ga(.(X2, X5), X7))
U27_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → QSORTF_IN_GA(.(X2, X5), X7)
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U31_GA(X1, X2, X3, X4, gtE_in_gg(X2, X1))
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → GTE_IN_GG(X2, X1)
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U32_GA(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U32_GA(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U33_GA(X1, X2, X3, X4, splitD_in_ggaa(X3, X1, X5, X6))
U32_GA(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → SPLITD_IN_GGAA(X3, X1, X5, X6)
U32_GA(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U34_GA(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U34_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U35_GA(X1, X2, X3, X4, qsortH_in_ga(X5, X7))
U34_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → QSORTH_IN_GA(X5, X7)
QSORTH_IN_GA(.(X1, X2), X3) → U10_GA(X1, X2, X3, splitD_in_ggaa(X2, X1, X4, X5))
QSORTH_IN_GA(.(X1, X2), X3) → SPLITD_IN_GGAA(X2, X1, X4, X5)
QSORTH_IN_GA(.(X1, X2), X3) → U11_GA(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U12_GA(X1, X2, X3, qsortH_in_ga(X4, X6))
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → QSORTH_IN_GA(X4, X6)
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U13_GA(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U13_GA(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U14_GA(X1, X2, X3, qsortH_in_ga(X5, X7))
U13_GA(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → QSORTH_IN_GA(X5, X7)
U13_GA(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U15_GA(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U15_GA(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U16_GA(X1, X2, X3, appendI_in_ggga(X6, X1, X7, X3))
U15_GA(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → APPENDI_IN_GGGA(X6, X1, X7, X3)
APPENDI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U17_GGGA(X1, X2, X3, X4, X5, appendI_in_ggga(X2, X3, X4, X5))
APPENDI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDI_IN_GGGA(X2, X3, X4, X5)
U34_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U36_GA(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U36_GA(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U37_GA(X1, X2, X3, X4, pG_in_gagga(.(X2, X6), X8, X7, X1, X4))
U36_GA(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → PG_IN_GAGGA(.(X2, X6), X8, X7, X1, X4)
PG_IN_GAGGA(X1, X2, X3, X4, X5) → U18_GAGGA(X1, X2, X3, X4, X5, qsortH_in_ga(X1, X2))
PG_IN_GAGGA(X1, X2, X3, X4, X5) → QSORTH_IN_GA(X1, X2)
PG_IN_GAGGA(X1, X2, X3, X4, X5) → U19_GAGGA(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U19_GAGGA(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U20_GAGGA(X1, X2, X3, X4, X5, appendB_in_ggga(X3, X4, X2, X5))
U19_GAGGA(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → APPENDB_IN_GGGA(X3, X4, X2, X5)
U27_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U29_GA(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
U29_GA(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U30_GA(X1, X2, X3, X4, pG_in_gagga(X6, X8, X7, X1, X4))
U29_GA(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → PG_IN_GAGGA(X6, X8, X7, X1, X4)

The TRS R consists of the following rules:

qsortcA_in_a([]) → qsortcA_out_a([])
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
qsortcF_in_ga([], []) → qsortcF_out_ga([], [])
qsortcF_in_ga(.(X1, []), X2) → U46_ga(X1, X2, qsortcA_in_a(X3))
U46_ga(X1, X2, qsortcA_out_a(X3)) → U47_ga(X1, X2, X3, qsortcA_in_a(X4))
U47_ga(X1, X2, X3, qsortcA_out_a(X4)) → U48_ga(X1, X2, appendcB_in_ggga(X3, X1, X4, X2))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U39_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U39_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U48_ga(X1, X2, appendcB_out_ggga(X3, X1, X4, X2)) → qsortcF_out_ga(.(X1, []), X2)
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U49_ga(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U49_ga(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U50_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U50_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U51_ga(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U53_ga(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U53_ga(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U54_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U54_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U55_ga(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U55_ga(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U56_ga(X1, X2, X3, X4, qcG_in_gagga(.(X2, X6), X8, X7, X1, X4))
qcG_in_gagga(X1, X2, X3, X4, X5) → U62_gagga(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U62_gagga(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U63_gagga(X1, X2, X3, X4, X5, appendcB_in_ggga(X3, X4, X2, X5))
U63_gagga(X1, X2, X3, X4, X5, appendcB_out_ggga(X3, X4, X2, X5)) → qcG_out_gagga(X1, X2, X3, X4, X5)
U56_ga(X1, X2, X3, X4, qcG_out_gagga(.(X2, X6), X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)
U51_ga(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U52_ga(X1, X2, X3, X4, qcG_in_gagga(X6, X8, X7, X1, X4))
U52_ga(X1, X2, X3, X4, qcG_out_gagga(X6, X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)

The argument filtering Pi contains the following mapping:
qsortF_in_ga(x1, x2) = qsortF_in_ga(x1)
.(x1, x2) = .(x1, x2)
[] = []
qsortcA_in_a(x1) = qsortcA_in_a
qsortcA_out_a(x1) = qsortcA_out_a(x1)
appendB_in_ggga(x1, x2, x3, x4) = appendB_in_ggga(x1, x2, x3)
leC_in_gg(x1, x2) = leC_in_gg(x1, x2)
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
splitD_in_ggaa(x1, x2, x3, x4) = splitD_in_ggaa(x1, x2)
gtE_in_gg(x1, x2) = gtE_in_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2)
splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_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)
qsortH_in_ga(x1, x2) = qsortH_in_ga(x1)
qsortcH_in_ga(x1, x2) = qsortcH_in_ga(x1)
qsortcH_out_ga(x1, x2) = qsortcH_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)
appendcI_in_ggga(x1, x2, x3, x4) = appendcI_in_ggga(x1, x2, x3)
appendcI_out_ggga(x1, x2, x3, x4) = appendcI_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
appendI_in_ggga(x1, x2, x3, x4) = appendI_in_ggga(x1, x2, x3)
pG_in_gagga(x1, x2, x3, x4, x5) = pG_in_gagga(x1, x3, x4)
qsortcF_in_ga(x1, x2) = qsortcF_in_ga(x1)
qsortcF_out_ga(x1, x2) = qsortcF_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)
appendcB_in_ggga(x1, x2, x3, x4) = appendcB_in_ggga(x1, x2, x3)
appendcB_out_ggga(x1, x2, x3, x4) = appendcB_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)
qcG_in_gagga(x1, x2, x3, x4, x5) = qcG_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)
qcG_out_gagga(x1, x2, x3, x4, x5) = qcG_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
QSORTF_IN_GA(x1, x2) = QSORTF_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)
APPENDB_IN_GGGA(x1, x2, x3, x4) = APPENDB_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)
LEC_IN_GG(x1, x2) = LEC_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)
SPLITD_IN_GGAA(x1, x2, x3, x4) = SPLITD_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)
GTE_IN_GG(x1, x2) = GTE_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)
QSORTH_IN_GA(x1, x2) = QSORTH_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)
APPENDI_IN_GGGA(x1, x2, x3, x4) = APPENDI_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)
PG_IN_GAGGA(x1, x2, x3, x4, x5) = PG_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:

QSORTF_IN_GA(.(X1, []), X2) → U21_GA(X1, X2, qsortcA_in_a(X3))
U21_GA(X1, X2, qsortcA_out_a(X3)) → U22_GA(X1, X2, X3, qsortcA_in_a(X4))
U22_GA(X1, X2, X3, qsortcA_out_a(X4)) → U23_GA(X1, X2, appendB_in_ggga(X3, X1, X4, X2))
U22_GA(X1, X2, X3, qsortcA_out_a(X4)) → APPENDB_IN_GGGA(X3, X1, X4, X2)
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U1_GGGA(X1, X2, X3, X4, X5, appendB_in_ggga(X2, X3, X4, X5))
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U24_GA(X1, X2, X3, X4, leC_in_gg(X2, X1))
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → LEC_IN_GG(X2, X1)
LEC_IN_GG(s(X1), s(X2)) → U2_GG(X1, X2, leC_in_gg(X1, X2))
LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U25_GA(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U25_GA(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U26_GA(X1, X2, X3, X4, splitD_in_ggaa(X3, X1, X5, X6))
U25_GA(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → SPLITD_IN_GGAA(X3, X1, X5, X6)
SPLITD_IN_GGAA(.(X1, X2), X3, .(X1, X4), X5) → U3_GGAA(X1, X2, X3, X4, X5, leC_in_gg(X1, X3))
SPLITD_IN_GGAA(.(X1, X2), X3, .(X1, X4), X5) → LEC_IN_GG(X1, X3)
SPLITD_IN_GGAA(.(X1, X2), X3, .(X1, X4), X5) → U4_GGAA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U4_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U5_GGAA(X1, X2, X3, X4, X5, splitD_in_ggaa(X2, X3, X4, X5))
U4_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3, X4, X5)
SPLITD_IN_GGAA(.(X1, X2), X3, X4, .(X1, X5)) → U6_GGAA(X1, X2, X3, X4, X5, gtE_in_gg(X1, X3))
SPLITD_IN_GGAA(.(X1, X2), X3, X4, .(X1, X5)) → GTE_IN_GG(X1, X3)
GTE_IN_GG(s(X1), s(X2)) → U9_GG(X1, X2, gtE_in_gg(X1, X2))
GTE_IN_GG(s(X1), s(X2)) → GTE_IN_GG(X1, X2)
SPLITD_IN_GGAA(.(X1, X2), X3, X4, .(X1, X5)) → U7_GGAA(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U7_GGAA(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U8_GGAA(X1, X2, X3, X4, X5, splitD_in_ggaa(X2, X3, X4, X5))
U7_GGAA(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3, X4, X5)
U25_GA(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U27_GA(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U27_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U28_GA(X1, X2, X3, X4, qsortF_in_ga(.(X2, X5), X7))
U27_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → QSORTF_IN_GA(.(X2, X5), X7)
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U31_GA(X1, X2, X3, X4, gtE_in_gg(X2, X1))
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → GTE_IN_GG(X2, X1)
QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U32_GA(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U32_GA(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U33_GA(X1, X2, X3, X4, splitD_in_ggaa(X3, X1, X5, X6))
U32_GA(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → SPLITD_IN_GGAA(X3, X1, X5, X6)
U32_GA(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U34_GA(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U34_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U35_GA(X1, X2, X3, X4, qsortH_in_ga(X5, X7))
U34_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → QSORTH_IN_GA(X5, X7)
QSORTH_IN_GA(.(X1, X2), X3) → U10_GA(X1, X2, X3, splitD_in_ggaa(X2, X1, X4, X5))
QSORTH_IN_GA(.(X1, X2), X3) → SPLITD_IN_GGAA(X2, X1, X4, X5)
QSORTH_IN_GA(.(X1, X2), X3) → U11_GA(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U12_GA(X1, X2, X3, qsortH_in_ga(X4, X6))
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → QSORTH_IN_GA(X4, X6)
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U13_GA(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U13_GA(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U14_GA(X1, X2, X3, qsortH_in_ga(X5, X7))
U13_GA(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → QSORTH_IN_GA(X5, X7)
U13_GA(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U15_GA(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U15_GA(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U16_GA(X1, X2, X3, appendI_in_ggga(X6, X1, X7, X3))
U15_GA(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → APPENDI_IN_GGGA(X6, X1, X7, X3)
APPENDI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U17_GGGA(X1, X2, X3, X4, X5, appendI_in_ggga(X2, X3, X4, X5))
APPENDI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDI_IN_GGGA(X2, X3, X4, X5)
U34_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U36_GA(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U36_GA(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U37_GA(X1, X2, X3, X4, pG_in_gagga(.(X2, X6), X8, X7, X1, X4))
U36_GA(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → PG_IN_GAGGA(.(X2, X6), X8, X7, X1, X4)
PG_IN_GAGGA(X1, X2, X3, X4, X5) → U18_GAGGA(X1, X2, X3, X4, X5, qsortH_in_ga(X1, X2))
PG_IN_GAGGA(X1, X2, X3, X4, X5) → QSORTH_IN_GA(X1, X2)
PG_IN_GAGGA(X1, X2, X3, X4, X5) → U19_GAGGA(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U19_GAGGA(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U20_GAGGA(X1, X2, X3, X4, X5, appendB_in_ggga(X3, X4, X2, X5))
U19_GAGGA(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → APPENDB_IN_GGGA(X3, X4, X2, X5)
U27_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U29_GA(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
U29_GA(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U30_GA(X1, X2, X3, X4, pG_in_gagga(X6, X8, X7, X1, X4))
U29_GA(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → PG_IN_GAGGA(X6, X8, X7, X1, X4)

The TRS R consists of the following rules:

qsortcA_in_a([]) → qsortcA_out_a([])
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
qsortcF_in_ga([], []) → qsortcF_out_ga([], [])
qsortcF_in_ga(.(X1, []), X2) → U46_ga(X1, X2, qsortcA_in_a(X3))
U46_ga(X1, X2, qsortcA_out_a(X3)) → U47_ga(X1, X2, X3, qsortcA_in_a(X4))
U47_ga(X1, X2, X3, qsortcA_out_a(X4)) → U48_ga(X1, X2, appendcB_in_ggga(X3, X1, X4, X2))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U39_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U39_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U48_ga(X1, X2, appendcB_out_ggga(X3, X1, X4, X2)) → qsortcF_out_ga(.(X1, []), X2)
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U49_ga(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U49_ga(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U50_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U50_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U51_ga(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U53_ga(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U53_ga(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U54_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U54_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U55_ga(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U55_ga(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U56_ga(X1, X2, X3, X4, qcG_in_gagga(.(X2, X6), X8, X7, X1, X4))
qcG_in_gagga(X1, X2, X3, X4, X5) → U62_gagga(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U62_gagga(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U63_gagga(X1, X2, X3, X4, X5, appendcB_in_ggga(X3, X4, X2, X5))
U63_gagga(X1, X2, X3, X4, X5, appendcB_out_ggga(X3, X4, X2, X5)) → qcG_out_gagga(X1, X2, X3, X4, X5)
U56_ga(X1, X2, X3, X4, qcG_out_gagga(.(X2, X6), X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)
U51_ga(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U52_ga(X1, X2, X3, X4, qcG_in_gagga(X6, X8, X7, X1, X4))
U52_ga(X1, X2, X3, X4, qcG_out_gagga(X6, X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)

(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:

APPENDI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDI_IN_GGGA(X2, X3, X4, X5)

The TRS R consists of the following rules:

qsortcA_in_a([]) → qsortcA_out_a([])
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
qsortcF_in_ga([], []) → qsortcF_out_ga([], [])
qsortcF_in_ga(.(X1, []), X2) → U46_ga(X1, X2, qsortcA_in_a(X3))
U46_ga(X1, X2, qsortcA_out_a(X3)) → U47_ga(X1, X2, X3, qsortcA_in_a(X4))
U47_ga(X1, X2, X3, qsortcA_out_a(X4)) → U48_ga(X1, X2, appendcB_in_ggga(X3, X1, X4, X2))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U39_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U39_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U48_ga(X1, X2, appendcB_out_ggga(X3, X1, X4, X2)) → qsortcF_out_ga(.(X1, []), X2)
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U49_ga(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U49_ga(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U50_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U50_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U51_ga(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U53_ga(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U53_ga(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U54_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U54_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U55_ga(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U55_ga(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U56_ga(X1, X2, X3, X4, qcG_in_gagga(.(X2, X6), X8, X7, X1, X4))
qcG_in_gagga(X1, X2, X3, X4, X5) → U62_gagga(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U62_gagga(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U63_gagga(X1, X2, X3, X4, X5, appendcB_in_ggga(X3, X4, X2, X5))
U63_gagga(X1, X2, X3, X4, X5, appendcB_out_ggga(X3, X4, X2, X5)) → qcG_out_gagga(X1, X2, X3, X4, X5)
U56_ga(X1, X2, X3, X4, qcG_out_gagga(.(X2, X6), X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)
U51_ga(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U52_ga(X1, X2, X3, X4, qcG_in_gagga(X6, X8, X7, X1, X4))
U52_ga(X1, X2, X3, X4, qcG_out_gagga(X6, X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
qsortcA_in_a(x1) = qsortcA_in_a
qsortcA_out_a(x1) = qsortcA_out_a(x1)
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2)
splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_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)
qsortcH_in_ga(x1, x2) = qsortcH_in_ga(x1)
qsortcH_out_ga(x1, x2) = qsortcH_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)
appendcI_in_ggga(x1, x2, x3, x4) = appendcI_in_ggga(x1, x2, x3)
appendcI_out_ggga(x1, x2, x3, x4) = appendcI_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
qsortcF_in_ga(x1, x2) = qsortcF_in_ga(x1)
qsortcF_out_ga(x1, x2) = qsortcF_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)
appendcB_in_ggga(x1, x2, x3, x4) = appendcB_in_ggga(x1, x2, x3)
appendcB_out_ggga(x1, x2, x3, x4) = appendcB_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)
qcG_in_gagga(x1, x2, x3, x4, x5) = qcG_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)
qcG_out_gagga(x1, x2, x3, x4, x5) = qcG_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
APPENDI_IN_GGGA(x1, x2, x3, x4) = APPENDI_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:

APPENDI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDI_IN_GGGA(X2, X3, X4, X5)

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

APPENDI_IN_GGGA(.(X1, X2), X3, X4) → APPENDI_IN_GGGA(X2, X3, X4)

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:

APPENDI_IN_GGGA(.(X1, X2), X3, X4) → APPENDI_IN_GGGA(X2, X3, X4)
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:

GTE_IN_GG(s(X1), s(X2)) → GTE_IN_GG(X1, X2)

The TRS R consists of the following rules:

qsortcA_in_a([]) → qsortcA_out_a([])
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
qsortcF_in_ga([], []) → qsortcF_out_ga([], [])
qsortcF_in_ga(.(X1, []), X2) → U46_ga(X1, X2, qsortcA_in_a(X3))
U46_ga(X1, X2, qsortcA_out_a(X3)) → U47_ga(X1, X2, X3, qsortcA_in_a(X4))
U47_ga(X1, X2, X3, qsortcA_out_a(X4)) → U48_ga(X1, X2, appendcB_in_ggga(X3, X1, X4, X2))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U39_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U39_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U48_ga(X1, X2, appendcB_out_ggga(X3, X1, X4, X2)) → qsortcF_out_ga(.(X1, []), X2)
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U49_ga(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U49_ga(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U50_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U50_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U51_ga(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U53_ga(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U53_ga(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U54_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U54_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U55_ga(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U55_ga(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U56_ga(X1, X2, X3, X4, qcG_in_gagga(.(X2, X6), X8, X7, X1, X4))
qcG_in_gagga(X1, X2, X3, X4, X5) → U62_gagga(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U62_gagga(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U63_gagga(X1, X2, X3, X4, X5, appendcB_in_ggga(X3, X4, X2, X5))
U63_gagga(X1, X2, X3, X4, X5, appendcB_out_ggga(X3, X4, X2, X5)) → qcG_out_gagga(X1, X2, X3, X4, X5)
U56_ga(X1, X2, X3, X4, qcG_out_gagga(.(X2, X6), X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)
U51_ga(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U52_ga(X1, X2, X3, X4, qcG_in_gagga(X6, X8, X7, X1, X4))
U52_ga(X1, X2, X3, X4, qcG_out_gagga(X6, X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
qsortcA_in_a(x1) = qsortcA_in_a
qsortcA_out_a(x1) = qsortcA_out_a(x1)
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2)
splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_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)
qsortcH_in_ga(x1, x2) = qsortcH_in_ga(x1)
qsortcH_out_ga(x1, x2) = qsortcH_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)
appendcI_in_ggga(x1, x2, x3, x4) = appendcI_in_ggga(x1, x2, x3)
appendcI_out_ggga(x1, x2, x3, x4) = appendcI_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
qsortcF_in_ga(x1, x2) = qsortcF_in_ga(x1)
qsortcF_out_ga(x1, x2) = qsortcF_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)
appendcB_in_ggga(x1, x2, x3, x4) = appendcB_in_ggga(x1, x2, x3)
appendcB_out_ggga(x1, x2, x3, x4) = appendcB_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)
qcG_in_gagga(x1, x2, x3, x4, x5) = qcG_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)
qcG_out_gagga(x1, x2, x3, x4, x5) = qcG_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
GTE_IN_GG(x1, x2) = GTE_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:

GTE_IN_GG(s(X1), s(X2)) → GTE_IN_GG(X1, X2)

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:

GTE_IN_GG(s(X1), s(X2)) → GTE_IN_GG(X1, X2)

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:

GTE_IN_GG(s(X1), s(X2)) → GTE_IN_GG(X1, X2)
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:

LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)

The TRS R consists of the following rules:

qsortcA_in_a([]) → qsortcA_out_a([])
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
qsortcF_in_ga([], []) → qsortcF_out_ga([], [])
qsortcF_in_ga(.(X1, []), X2) → U46_ga(X1, X2, qsortcA_in_a(X3))
U46_ga(X1, X2, qsortcA_out_a(X3)) → U47_ga(X1, X2, X3, qsortcA_in_a(X4))
U47_ga(X1, X2, X3, qsortcA_out_a(X4)) → U48_ga(X1, X2, appendcB_in_ggga(X3, X1, X4, X2))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U39_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U39_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U48_ga(X1, X2, appendcB_out_ggga(X3, X1, X4, X2)) → qsortcF_out_ga(.(X1, []), X2)
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U49_ga(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U49_ga(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U50_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U50_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U51_ga(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U53_ga(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U53_ga(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U54_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U54_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U55_ga(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U55_ga(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U56_ga(X1, X2, X3, X4, qcG_in_gagga(.(X2, X6), X8, X7, X1, X4))
qcG_in_gagga(X1, X2, X3, X4, X5) → U62_gagga(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U62_gagga(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U63_gagga(X1, X2, X3, X4, X5, appendcB_in_ggga(X3, X4, X2, X5))
U63_gagga(X1, X2, X3, X4, X5, appendcB_out_ggga(X3, X4, X2, X5)) → qcG_out_gagga(X1, X2, X3, X4, X5)
U56_ga(X1, X2, X3, X4, qcG_out_gagga(.(X2, X6), X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)
U51_ga(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U52_ga(X1, X2, X3, X4, qcG_in_gagga(X6, X8, X7, X1, X4))
U52_ga(X1, X2, X3, X4, qcG_out_gagga(X6, X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
qsortcA_in_a(x1) = qsortcA_in_a
qsortcA_out_a(x1) = qsortcA_out_a(x1)
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2)
splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_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)
qsortcH_in_ga(x1, x2) = qsortcH_in_ga(x1)
qsortcH_out_ga(x1, x2) = qsortcH_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)
appendcI_in_ggga(x1, x2, x3, x4) = appendcI_in_ggga(x1, x2, x3)
appendcI_out_ggga(x1, x2, x3, x4) = appendcI_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
qsortcF_in_ga(x1, x2) = qsortcF_in_ga(x1)
qsortcF_out_ga(x1, x2) = qsortcF_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)
appendcB_in_ggga(x1, x2, x3, x4) = appendcB_in_ggga(x1, x2, x3)
appendcB_out_ggga(x1, x2, x3, x4) = appendcB_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)
qcG_in_gagga(x1, x2, x3, x4, x5) = qcG_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)
qcG_out_gagga(x1, x2, x3, x4, x5) = qcG_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
LEC_IN_GG(x1, x2) = LEC_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:

LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)

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:

LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)

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:

LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)
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:

SPLITD_IN_GGAA(.(X1, X2), X3, .(X1, X4), X5) → U4_GGAA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U4_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3, X4, X5)
SPLITD_IN_GGAA(.(X1, X2), X3, X4, .(X1, X5)) → U7_GGAA(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U7_GGAA(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3, X4, X5)

The TRS R consists of the following rules:

qsortcA_in_a([]) → qsortcA_out_a([])
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
qsortcF_in_ga([], []) → qsortcF_out_ga([], [])
qsortcF_in_ga(.(X1, []), X2) → U46_ga(X1, X2, qsortcA_in_a(X3))
U46_ga(X1, X2, qsortcA_out_a(X3)) → U47_ga(X1, X2, X3, qsortcA_in_a(X4))
U47_ga(X1, X2, X3, qsortcA_out_a(X4)) → U48_ga(X1, X2, appendcB_in_ggga(X3, X1, X4, X2))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U39_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U39_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U48_ga(X1, X2, appendcB_out_ggga(X3, X1, X4, X2)) → qsortcF_out_ga(.(X1, []), X2)
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U49_ga(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U49_ga(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U50_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U50_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U51_ga(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U53_ga(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U53_ga(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U54_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U54_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U55_ga(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U55_ga(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U56_ga(X1, X2, X3, X4, qcG_in_gagga(.(X2, X6), X8, X7, X1, X4))
qcG_in_gagga(X1, X2, X3, X4, X5) → U62_gagga(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U62_gagga(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U63_gagga(X1, X2, X3, X4, X5, appendcB_in_ggga(X3, X4, X2, X5))
U63_gagga(X1, X2, X3, X4, X5, appendcB_out_ggga(X3, X4, X2, X5)) → qcG_out_gagga(X1, X2, X3, X4, X5)
U56_ga(X1, X2, X3, X4, qcG_out_gagga(.(X2, X6), X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)
U51_ga(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U52_ga(X1, X2, X3, X4, qcG_in_gagga(X6, X8, X7, X1, X4))
U52_ga(X1, X2, X3, X4, qcG_out_gagga(X6, X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
qsortcA_in_a(x1) = qsortcA_in_a
qsortcA_out_a(x1) = qsortcA_out_a(x1)
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2)
splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_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)
qsortcH_in_ga(x1, x2) = qsortcH_in_ga(x1)
qsortcH_out_ga(x1, x2) = qsortcH_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)
appendcI_in_ggga(x1, x2, x3, x4) = appendcI_in_ggga(x1, x2, x3)
appendcI_out_ggga(x1, x2, x3, x4) = appendcI_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
qsortcF_in_ga(x1, x2) = qsortcF_in_ga(x1)
qsortcF_out_ga(x1, x2) = qsortcF_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)
appendcB_in_ggga(x1, x2, x3, x4) = appendcB_in_ggga(x1, x2, x3)
appendcB_out_ggga(x1, x2, x3, x4) = appendcB_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)
qcG_in_gagga(x1, x2, x3, x4, x5) = qcG_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)
qcG_out_gagga(x1, x2, x3, x4, x5) = qcG_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
SPLITD_IN_GGAA(x1, x2, x3, x4) = SPLITD_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:

SPLITD_IN_GGAA(.(X1, X2), X3, .(X1, X4), X5) → U4_GGAA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U4_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3, X4, X5)
SPLITD_IN_GGAA(.(X1, X2), X3, X4, .(X1, X5)) → U7_GGAA(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U7_GGAA(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3, X4, X5)

The TRS R consists of the following rules:

lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
SPLITD_IN_GGAA(x1, x2, x3, x4) = SPLITD_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:

SPLITD_IN_GGAA(.(X1, X2), X3) → U4_GGAA(X1, X2, X3, lecC_in_gg(X1, X3))
U4_GGAA(X1, X2, X3, lecC_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3)
SPLITD_IN_GGAA(.(X1, X2), X3) → U7_GGAA(X1, X2, X3, gtcE_in_gg(X1, X3))
U7_GGAA(X1, X2, X3, gtcE_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3)

The TRS R consists of the following rules:

lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecC_in_gg(x0, x1)
gtcE_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(X1, X2, X3, lecC_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3)
The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2
U7_GGAA(X1, X2, X3, gtcE_out_gg(X1, X3)) → SPLITD_IN_GGAA(X2, X3)
The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2
SPLITD_IN_GGAA(.(X1, X2), X3) → U4_GGAA(X1, X2, X3, lecC_in_gg(X1, X3))
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3
SPLITD_IN_GGAA(.(X1, X2), X3) → U7_GGAA(X1, X2, X3, gtcE_in_gg(X1, X3))
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:

QSORTH_IN_GA(.(X1, X2), X3) → U11_GA(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → QSORTH_IN_GA(X4, X6)
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U13_GA(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U13_GA(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → QSORTH_IN_GA(X5, X7)

The TRS R consists of the following rules:

qsortcA_in_a([]) → qsortcA_out_a([])
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
qsortcF_in_ga([], []) → qsortcF_out_ga([], [])
qsortcF_in_ga(.(X1, []), X2) → U46_ga(X1, X2, qsortcA_in_a(X3))
U46_ga(X1, X2, qsortcA_out_a(X3)) → U47_ga(X1, X2, X3, qsortcA_in_a(X4))
U47_ga(X1, X2, X3, qsortcA_out_a(X4)) → U48_ga(X1, X2, appendcB_in_ggga(X3, X1, X4, X2))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U39_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U39_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U48_ga(X1, X2, appendcB_out_ggga(X3, X1, X4, X2)) → qsortcF_out_ga(.(X1, []), X2)
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U49_ga(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U49_ga(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U50_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U50_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U51_ga(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U53_ga(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U53_ga(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U54_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U54_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U55_ga(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U55_ga(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U56_ga(X1, X2, X3, X4, qcG_in_gagga(.(X2, X6), X8, X7, X1, X4))
qcG_in_gagga(X1, X2, X3, X4, X5) → U62_gagga(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U62_gagga(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U63_gagga(X1, X2, X3, X4, X5, appendcB_in_ggga(X3, X4, X2, X5))
U63_gagga(X1, X2, X3, X4, X5, appendcB_out_ggga(X3, X4, X2, X5)) → qcG_out_gagga(X1, X2, X3, X4, X5)
U56_ga(X1, X2, X3, X4, qcG_out_gagga(.(X2, X6), X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)
U51_ga(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U52_ga(X1, X2, X3, X4, qcG_in_gagga(X6, X8, X7, X1, X4))
U52_ga(X1, X2, X3, X4, qcG_out_gagga(X6, X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
qsortcA_in_a(x1) = qsortcA_in_a
qsortcA_out_a(x1) = qsortcA_out_a(x1)
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2)
splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_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)
qsortcH_in_ga(x1, x2) = qsortcH_in_ga(x1)
qsortcH_out_ga(x1, x2) = qsortcH_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)
appendcI_in_ggga(x1, x2, x3, x4) = appendcI_in_ggga(x1, x2, x3)
appendcI_out_ggga(x1, x2, x3, x4) = appendcI_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
qsortcF_in_ga(x1, x2) = qsortcF_in_ga(x1)
qsortcF_out_ga(x1, x2) = qsortcF_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)
appendcB_in_ggga(x1, x2, x3, x4) = appendcB_in_ggga(x1, x2, x3)
appendcB_out_ggga(x1, x2, x3, x4) = appendcB_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)
qcG_in_gagga(x1, x2, x3, x4, x5) = qcG_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)
qcG_out_gagga(x1, x2, x3, x4, x5) = qcG_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
QSORTH_IN_GA(x1, x2) = QSORTH_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:

QSORTH_IN_GA(.(X1, X2), X3) → U11_GA(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → QSORTH_IN_GA(X4, X6)
U11_GA(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U13_GA(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U13_GA(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → QSORTH_IN_GA(X5, X7)

The TRS R consists of the following rules:

splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2)
splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_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)
qsortcH_in_ga(x1, x2) = qsortcH_in_ga(x1)
qsortcH_out_ga(x1, x2) = qsortcH_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)
appendcI_in_ggga(x1, x2, x3, x4) = appendcI_in_ggga(x1, x2, x3)
appendcI_out_ggga(x1, x2, x3, x4) = appendcI_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
QSORTH_IN_GA(x1, x2) = QSORTH_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:

QSORTH_IN_GA(.(X1, X2)) → U11_GA(X1, X2, splitcD_in_ggaa(X2, X1))
U11_GA(X1, X2, splitcD_out_ggaa(X2, X1, X4, X5)) → QSORTH_IN_GA(X4)
U11_GA(X1, X2, splitcD_out_ggaa(X2, X1, X4, X5)) → U13_GA(X1, X2, X5, qsortcH_in_ga(X4))
U13_GA(X1, X2, X5, qsortcH_out_ga(X4, X6)) → QSORTH_IN_GA(X5)

The TRS R consists of the following rules:

splitcD_in_ggaa([], X1) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3) → U41_ggaa(X1, X2, X3, lecC_in_gg(X1, X3))
splitcD_in_ggaa(.(X1, X2), X3) → U43_ggaa(X1, X2, X3, gtcE_in_gg(X1, X3))
qsortcH_in_ga([]) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2)) → U57_ga(X1, X2, splitcD_in_ggaa(X2, X1))
U41_ggaa(X1, X2, X3, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U43_ggaa(X1, X2, X3, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U57_ga(X1, X2, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X5, qsortcH_in_ga(X4))
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U42_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U44_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U58_ga(X1, X2, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X6, qsortcH_in_ga(X5))
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
U59_ga(X1, X2, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, appendcI_in_ggga(X6, X1, X7))
U60_ga(X1, X2, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
appendcI_in_ggga([], X1, X2) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4) → U61_ggga(X1, X2, X3, X4, appendcI_in_ggga(X2, X3, X4))
U61_ggga(X1, X2, X3, X4, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

splitcD_in_ggaa(x0, x1)
qsortcH_in_ga(x0)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U57_ga(x0, x1, x2)
lecC_in_gg(x0, x1)
U42_ggaa(x0, x1, x2, x3)
gtcE_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)
appendcI_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,JAR06].

The following pairs can be oriented strictly and are deleted.

QSORTH_IN_GA(.(X1, X2)) → U11_GA(X1, X2, splitcD_in_ggaa(X2, X1))

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

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(QSORTH_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(appendcI_in_ggga(x₁, x₂, x₃)) = 0
POL(appendcI_out_ggga(x₁, x₂, x₃, x₄)) = 0
POL(gtcE_in_gg(x₁, x₂)) = 0
POL(gtcE_out_gg(x₁, x₂)) = 0
POL(lecC_in_gg(x₁, x₂)) = 0
POL(lecC_out_gg(x₁, x₂)) = 0
POL(qsortcH_in_ga(x₁)) = 0
POL(qsortcH_out_ga(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(splitcD_in_ggaa(x₁, x₂)) = x₁
POL(splitcD_out_ggaa(x₁, x₂, x₃, x₄)) = x₃ + x₄

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

splitcD_in_ggaa([], X1) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3) → U41_ggaa(X1, X2, X3, lecC_in_gg(X1, X3))
splitcD_in_ggaa(.(X1, X2), X3) → U43_ggaa(X1, X2, X3, gtcE_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U42_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
U43_ggaa(X1, X2, X3, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U44_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))

(43) Obligation:

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

U11_GA(X1, X2, splitcD_out_ggaa(X2, X1, X4, X5)) → QSORTH_IN_GA(X4)
U11_GA(X1, X2, splitcD_out_ggaa(X2, X1, X4, X5)) → U13_GA(X1, X2, X5, qsortcH_in_ga(X4))
U13_GA(X1, X2, X5, qsortcH_out_ga(X4, X6)) → QSORTH_IN_GA(X5)

The TRS R consists of the following rules:

splitcD_in_ggaa([], X1) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3) → U41_ggaa(X1, X2, X3, lecC_in_gg(X1, X3))
splitcD_in_ggaa(.(X1, X2), X3) → U43_ggaa(X1, X2, X3, gtcE_in_gg(X1, X3))
qsortcH_in_ga([]) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2)) → U57_ga(X1, X2, splitcD_in_ggaa(X2, X1))
U41_ggaa(X1, X2, X3, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U43_ggaa(X1, X2, X3, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U57_ga(X1, X2, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X5, qsortcH_in_ga(X4))
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U42_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U44_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U58_ga(X1, X2, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X6, qsortcH_in_ga(X5))
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
U59_ga(X1, X2, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, appendcI_in_ggga(X6, X1, X7))
U60_ga(X1, X2, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
appendcI_in_ggga([], X1, X2) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4) → U61_ggga(X1, X2, X3, X4, appendcI_in_ggga(X2, X3, X4))
U61_ggga(X1, X2, X3, X4, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

splitcD_in_ggaa(x0, x1)
qsortcH_in_ga(x0)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U57_ga(x0, x1, x2)
lecC_in_gg(x0, x1)
U42_ggaa(x0, x1, x2, x3)
gtcE_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)
appendcI_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:

APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)

The TRS R consists of the following rules:

qsortcA_in_a([]) → qsortcA_out_a([])
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
qsortcF_in_ga([], []) → qsortcF_out_ga([], [])
qsortcF_in_ga(.(X1, []), X2) → U46_ga(X1, X2, qsortcA_in_a(X3))
U46_ga(X1, X2, qsortcA_out_a(X3)) → U47_ga(X1, X2, X3, qsortcA_in_a(X4))
U47_ga(X1, X2, X3, qsortcA_out_a(X4)) → U48_ga(X1, X2, appendcB_in_ggga(X3, X1, X4, X2))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U39_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U39_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U48_ga(X1, X2, appendcB_out_ggga(X3, X1, X4, X2)) → qsortcF_out_ga(.(X1, []), X2)
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U49_ga(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U49_ga(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U50_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U50_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U51_ga(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U53_ga(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U53_ga(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U54_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U54_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U55_ga(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U55_ga(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U56_ga(X1, X2, X3, X4, qcG_in_gagga(.(X2, X6), X8, X7, X1, X4))
qcG_in_gagga(X1, X2, X3, X4, X5) → U62_gagga(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U62_gagga(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U63_gagga(X1, X2, X3, X4, X5, appendcB_in_ggga(X3, X4, X2, X5))
U63_gagga(X1, X2, X3, X4, X5, appendcB_out_ggga(X3, X4, X2, X5)) → qcG_out_gagga(X1, X2, X3, X4, X5)
U56_ga(X1, X2, X3, X4, qcG_out_gagga(.(X2, X6), X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)
U51_ga(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U52_ga(X1, X2, X3, X4, qcG_in_gagga(X6, X8, X7, X1, X4))
U52_ga(X1, X2, X3, X4, qcG_out_gagga(X6, X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
qsortcA_in_a(x1) = qsortcA_in_a
qsortcA_out_a(x1) = qsortcA_out_a(x1)
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2)
splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_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)
qsortcH_in_ga(x1, x2) = qsortcH_in_ga(x1)
qsortcH_out_ga(x1, x2) = qsortcH_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)
appendcI_in_ggga(x1, x2, x3, x4) = appendcI_in_ggga(x1, x2, x3)
appendcI_out_ggga(x1, x2, x3, x4) = appendcI_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
qsortcF_in_ga(x1, x2) = qsortcF_in_ga(x1)
qsortcF_out_ga(x1, x2) = qsortcF_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)
appendcB_in_ggga(x1, x2, x3, x4) = appendcB_in_ggga(x1, x2, x3)
appendcB_out_ggga(x1, x2, x3, x4) = appendcB_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)
qcG_in_gagga(x1, x2, x3, x4, x5) = qcG_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)
qcG_out_gagga(x1, x2, x3, x4, x5) = qcG_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
APPENDB_IN_GGGA(x1, x2, x3, x4) = APPENDB_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:

APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)

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

APPENDB_IN_GGGA(.(X1, X2), X3, X4) → APPENDB_IN_GGGA(X2, X3, X4)

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:

APPENDB_IN_GGGA(.(X1, X2), X3, X4) → APPENDB_IN_GGGA(X2, X3, X4)
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:

QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U25_GA(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U25_GA(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U27_GA(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U27_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → QSORTF_IN_GA(.(X2, X5), X7)

The TRS R consists of the following rules:

qsortcA_in_a([]) → qsortcA_out_a([])
lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
qsortcH_in_ga([], []) → qsortcH_out_ga([], [])
qsortcH_in_ga(.(X1, X2), X3) → U57_ga(X1, X2, X3, splitcD_in_ggaa(X2, X1, X4, X5))
U57_ga(X1, X2, X3, splitcD_out_ggaa(X2, X1, X4, X5)) → U58_ga(X1, X2, X3, X5, qsortcH_in_ga(X4, X6))
U58_ga(X1, X2, X3, X5, qsortcH_out_ga(X4, X6)) → U59_ga(X1, X2, X3, X6, qsortcH_in_ga(X5, X7))
U59_ga(X1, X2, X3, X6, qsortcH_out_ga(X5, X7)) → U60_ga(X1, X2, X3, appendcI_in_ggga(X6, X1, X7, X3))
appendcI_in_ggga([], X1, X2, .(X1, X2)) → appendcI_out_ggga([], X1, X2, .(X1, X2))
appendcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U61_ggga(X1, X2, X3, X4, X5, appendcI_in_ggga(X2, X3, X4, X5))
U61_ggga(X1, X2, X3, X4, X5, appendcI_out_ggga(X2, X3, X4, X5)) → appendcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U60_ga(X1, X2, X3, appendcI_out_ggga(X6, X1, X7, X3)) → qsortcH_out_ga(.(X1, X2), X3)
qsortcF_in_ga([], []) → qsortcF_out_ga([], [])
qsortcF_in_ga(.(X1, []), X2) → U46_ga(X1, X2, qsortcA_in_a(X3))
U46_ga(X1, X2, qsortcA_out_a(X3)) → U47_ga(X1, X2, X3, qsortcA_in_a(X4))
U47_ga(X1, X2, X3, qsortcA_out_a(X4)) → U48_ga(X1, X2, appendcB_in_ggga(X3, X1, X4, X2))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U39_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U39_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U48_ga(X1, X2, appendcB_out_ggga(X3, X1, X4, X2)) → qsortcF_out_ga(.(X1, []), X2)
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U49_ga(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U49_ga(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U50_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U50_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U51_ga(X1, X2, X3, X4, X6, qsortcF_in_ga(.(X2, X5), X7))
qsortcF_in_ga(.(X1, .(X2, X3)), X4) → U53_ga(X1, X2, X3, X4, gtcE_in_gg(X2, X1))
U53_ga(X1, X2, X3, X4, gtcE_out_gg(X2, X1)) → U54_ga(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U54_ga(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → U55_ga(X1, X2, X3, X4, X6, qsortcH_in_ga(X5, X7))
U55_ga(X1, X2, X3, X4, X6, qsortcH_out_ga(X5, X7)) → U56_ga(X1, X2, X3, X4, qcG_in_gagga(.(X2, X6), X8, X7, X1, X4))
qcG_in_gagga(X1, X2, X3, X4, X5) → U62_gagga(X1, X2, X3, X4, X5, qsortcH_in_ga(X1, X2))
U62_gagga(X1, X2, X3, X4, X5, qsortcH_out_ga(X1, X2)) → U63_gagga(X1, X2, X3, X4, X5, appendcB_in_ggga(X3, X4, X2, X5))
U63_gagga(X1, X2, X3, X4, X5, appendcB_out_ggga(X3, X4, X2, X5)) → qcG_out_gagga(X1, X2, X3, X4, X5)
U56_ga(X1, X2, X3, X4, qcG_out_gagga(.(X2, X6), X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)
U51_ga(X1, X2, X3, X4, X6, qsortcF_out_ga(.(X2, X5), X7)) → U52_ga(X1, X2, X3, X4, qcG_in_gagga(X6, X8, X7, X1, X4))
U52_ga(X1, X2, X3, X4, qcG_out_gagga(X6, X8, X7, X1, X4)) → qsortcF_out_ga(.(X1, .(X2, X3)), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
qsortcA_in_a(x1) = qsortcA_in_a
qsortcA_out_a(x1) = qsortcA_out_a(x1)
s(x1) = s(x1)
lecC_in_gg(x1, x2) = lecC_in_gg(x1, x2)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lecC_out_gg(x1, x2) = lecC_out_gg(x1, x2)
gtcE_in_gg(x1, x2) = gtcE_in_gg(x1, x2)
U45_gg(x1, x2, x3) = U45_gg(x1, x2, x3)
gtcE_out_gg(x1, x2) = gtcE_out_gg(x1, x2)
splitcD_in_ggaa(x1, x2, x3, x4) = splitcD_in_ggaa(x1, x2)
splitcD_out_ggaa(x1, x2, x3, x4) = splitcD_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)
qsortcH_in_ga(x1, x2) = qsortcH_in_ga(x1)
qsortcH_out_ga(x1, x2) = qsortcH_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)
appendcI_in_ggga(x1, x2, x3, x4) = appendcI_in_ggga(x1, x2, x3)
appendcI_out_ggga(x1, x2, x3, x4) = appendcI_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6) = U61_ggga(x1, x2, x3, x4, x6)
qsortcF_in_ga(x1, x2) = qsortcF_in_ga(x1)
qsortcF_out_ga(x1, x2) = qsortcF_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)
appendcB_in_ggga(x1, x2, x3, x4) = appendcB_in_ggga(x1, x2, x3)
appendcB_out_ggga(x1, x2, x3, x4) = appendcB_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)
qcG_in_gagga(x1, x2, x3, x4, x5) = qcG_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)
qcG_out_gagga(x1, x2, x3, x4, x5) = qcG_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5) = U52_ga(x1, x2, x3, x5)
QSORTF_IN_GA(x1, x2) = QSORTF_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:

QSORTF_IN_GA(.(X1, .(X2, X3)), X4) → U25_GA(X1, X2, X3, X4, lecC_in_gg(X2, X1))
U25_GA(X1, X2, X3, X4, lecC_out_gg(X2, X1)) → U27_GA(X1, X2, X3, X4, splitcD_in_ggaa(X3, X1, X5, X6))
U27_GA(X1, X2, X3, X4, splitcD_out_ggaa(X3, X1, X5, X6)) → QSORTF_IN_GA(.(X2, X5), X7)

The TRS R consists of the following rules:

lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
splitcD_in_ggaa([], X1, [], []) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3, .(X1, X4), X5) → U41_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
splitcD_in_ggaa(.(X1, X2), X3, X4, .(X1, X5)) → U43_ggaa(X1, X2, X3, X4, X5, gtcE_in_gg(X1, X3))
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U41_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U43_ggaa(X1, X2, X3, X4, X5, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, X4, X5, splitcD_in_ggaa(X2, X3, X4, X5))
U42_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U44_ggaa(X1, X2, X3, X4, X5, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))

(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:

QSORTF_IN_GA(.(X1, .(X2, X3))) → U25_GA(X1, X2, X3, lecC_in_gg(X2, X1))
U25_GA(X1, X2, X3, lecC_out_gg(X2, X1)) → U27_GA(X1, X2, X3, splitcD_in_ggaa(X3, X1))
U27_GA(X1, X2, X3, splitcD_out_ggaa(X3, X1, X5, X6)) → QSORTF_IN_GA(.(X2, X5))

The TRS R consists of the following rules:

lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
splitcD_in_ggaa([], X1) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3) → U41_ggaa(X1, X2, X3, lecC_in_gg(X1, X3))
splitcD_in_ggaa(.(X1, X2), X3) → U43_ggaa(X1, X2, X3, gtcE_in_gg(X1, X3))
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U41_ggaa(X1, X2, X3, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U43_ggaa(X1, X2, X3, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U42_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U44_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecC_in_gg(x0, x1)
splitcD_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)
gtcE_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,JAR06].

The following pairs can be oriented strictly and are deleted.

QSORTF_IN_GA(.(X1, .(X2, X3))) → U25_GA(X1, X2, X3, lecC_in_gg(X2, X1))

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

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(QSORTF_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(gtcE_in_gg(x₁, x₂)) = 0
POL(gtcE_out_gg(x₁, x₂)) = 0
POL(lecC_in_gg(x₁, x₂)) = 0
POL(lecC_out_gg(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(splitcD_in_ggaa(x₁, x₂)) = x₁
POL(splitcD_out_ggaa(x₁, x₂, x₃, x₄)) = x₃

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

splitcD_in_ggaa([], X1) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3) → U41_ggaa(X1, X2, X3, lecC_in_gg(X1, X3))
splitcD_in_ggaa(.(X1, X2), X3) → U43_ggaa(X1, X2, X3, gtcE_in_gg(X1, X3))
U41_ggaa(X1, X2, X3, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U42_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
U43_ggaa(X1, X2, X3, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U44_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))

(59) Obligation:

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

U25_GA(X1, X2, X3, lecC_out_gg(X2, X1)) → U27_GA(X1, X2, X3, splitcD_in_ggaa(X3, X1))
U27_GA(X1, X2, X3, splitcD_out_ggaa(X3, X1, X5, X6)) → QSORTF_IN_GA(.(X2, X5))

The TRS R consists of the following rules:

lecC_in_gg(s(X1), s(X2)) → U40_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
splitcD_in_ggaa([], X1) → splitcD_out_ggaa([], X1, [], [])
splitcD_in_ggaa(.(X1, X2), X3) → U41_ggaa(X1, X2, X3, lecC_in_gg(X1, X3))
splitcD_in_ggaa(.(X1, X2), X3) → U43_ggaa(X1, X2, X3, gtcE_in_gg(X1, X3))
U40_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U41_ggaa(X1, X2, X3, lecC_out_gg(X1, X3)) → U42_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U43_ggaa(X1, X2, X3, gtcE_out_gg(X1, X3)) → U44_ggaa(X1, X2, X3, splitcD_in_ggaa(X2, X3))
U42_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, .(X1, X4), X5)
gtcE_in_gg(s(X1), s(X2)) → U45_gg(X1, X2, gtcE_in_gg(X1, X2))
gtcE_in_gg(s(X1), 0) → gtcE_out_gg(s(X1), 0)
U44_ggaa(X1, X2, X3, splitcD_out_ggaa(X2, X3, X4, X5)) → splitcD_out_ggaa(.(X1, X2), X3, X4, .(X1, X5))
U45_gg(X1, X2, gtcE_out_gg(X1, X2)) → gtcE_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecC_in_gg(x0, x1)
splitcD_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)
gtcE_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.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

(4) Obligation:

(5) TriplesToPiDPProof (SOUND transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (EQUIVALENT transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) Obligation:

(31) UsableRulesProof (EQUIVALENT transformation)

(32) Obligation:

(33) PiDPToQDPProof (SOUND transformation)

(34) Obligation:

(35) QDPSizeChangeProof (EQUIVALENT transformation)

(36) YES

(37) Obligation:

(38) UsableRulesProof (EQUIVALENT transformation)

(39) Obligation:

(40) PiDPToQDPProof (SOUND transformation)

(41) Obligation:

(42) QDPOrderProof (EQUIVALENT transformation)

(43) Obligation:

(44) DependencyGraphProof (EQUIVALENT transformation)

(45) TRUE

(46) Obligation:

(47) UsableRulesProof (EQUIVALENT transformation)

(48) Obligation:

(49) PiDPToQDPProof (SOUND transformation)

(50) Obligation:

(51) QDPSizeChangeProof (EQUIVALENT transformation)

(52) YES

(53) Obligation:

(54) UsableRulesProof (EQUIVALENT transformation)

(55) Obligation:

(56) PiDPToQDPProof (SOUND transformation)

(57) Obligation:

(58) QDPOrderProof (EQUIVALENT transformation)

(59) Obligation:

(60) DependencyGraphProof (EQUIVALENT transformation)

(61) TRUE